TOP PRIORITY
The following are things I think are absolutely necessary for 0.9.9.9:
*Make most special levels on Nightmare require killing every distinct enemy once to clear them rather than having every enemy dead at once, so that they no longer require an extremely tedious ordeal of corpse disposal to clear (especially now that the Enraged Timer could make it near impossible to clear the earlier special levels on N!). Additionally rebalance their enemy loadouts to incorporate the new enemies added, particularly on Nightmare (go to this post for details).
*Finish the trait rebalancing since we got some loose ends there still (covered in this post, particularly Bullet Dance still needs its buff, Survivalist + Entrenchment are overtuned, and Tough As Nails could be toned down to the originally proposed +5% energy resistances per level). The masteries can all be made less restrictive too as covered in this post.
*Fix the error with selecting Bullet Dance, and the bug with additive traits that sometimes causes their bonuses to get applied twice in a level up.
*Patch up the rough edges of infighting, as covered in this post (namely, add a flag that disables infighting in boss levels and certain special levels, disable infighting when Enraged Mode is triggered or with hunting enemies, don't have enemies infight their own species, don't let enemies retaliate against Archviles, and if possible, prevent enemies in a spawned monster group from infighting each other). There is also some remaining AI issues in that post that need to be fixed too (namely, Formers/Nobles will occasionally get stuck on items/armor instead of picking it up, Formers/Nobles will sometimes not use items at low health for unclear reasons, Lost Souls/Elementals/Cyberdemons will not aggro when you hit them out of their vision, and Souls/Cyberdemons will not retaliate when hit by other enemies).
*The remaining bugs and issues in this post that have not been addressed yet (too many to list here, just CTRL + F "Grappling Armor" and look at every entry from there).
*Fix the unwinnable 4 Angelic Badges so that true 100% completion can be possible; UAC Angelic, Demonic Angelic, Speedrunner Angelic, and Pacifist Angelic (go to the Badges post of the OP for current suggestions on what to do with them).
*Fix Item Reveal not working in Nightmare when all enemies are dead. Also add a new message that notifies you when all distinct enemies on a floor have been killed in Nightmare and Angel of Darkness, so that players seeking 100% kills don't need to keep bringing up the Player Info screen to check (and perhaps tie the aforementioned Item Reveal to triggering that message?).
*Replace the redundant Missile Launcher in City Of Skulls/Abyssal Plains (current suggestion is a guaranteed randomized Exotic Armor since current Deimos special levels are lacking in armor rewards).
*Fix the softlocks in The Wall, Containment Area, and Mt. Erebus
HIGH PRIORITY
These are important things I think would really improve 0.9.9.9 and a good chunk of should make it into 0.9.9.9, but they aren't all necessary to implement now if stuff has to be left on the wayside for the next version (1.0.0.0 or 0.9.9.9.1?):
*The Chaingun, Frag Shotgun, Assault Rifle, and mods changes listed in the equipment section of the OP (the other suggested equipment changes listed would be nice, but those ones aren't as important and so I would put them down in the mid priority tier).
*Give default sounds to all mute enemies (the Nightmare enemies, Elite Formers, Lava Elemental, Agony Elemental, Arena Master, and Apostle). Aside from the subjective value of it livening up the game, it eliminates the ethical grey area with players modding sounds for those enemies, and it would be nice compensation for the Enraged Timer no longer letting you camp them too, so that players exploring around without Intuition 2 can detect these enemies are around without stumbling face first into them. Since adding new sounds into the game isn't on the table because of copyright concerns, Nightmare enemies and Elite Formers can just reuse the sounds of their normal counterparts, the Agony Elemental can just reuse the sounds of the Pain Elemental, and the Arena Master and Apostle can just reuse the sounds of the Archvile; Lava Elemental is unclear with what sounds to give, though in my own personal setup I gave it Mancubus sounds.
*Nerf the amount of bullets that Former Captains carry to 40, and buff the Nightmare Archvile's explosion radius to 2, as covered in the enemy section on the OP. Also Nightmare Cacodemons are overtuned after their massive buff and can be a bit tedious to kill, so they should have their current 50% fire and acid resistances nerfed or removed outright, or alternatively, have their innate armor reduced to 1 or give them a -25% resistance to bullet + sharpnel damage to better emphasize their vulnerability to physical weapons.
*Make single enemy type levels be able to have Nightmare enemies just like how Caves can (and make Archvile + Elite Formers be possible in the Black Magic level type) at sufficiently high level rolls, so that they no longer remain breather levels in deeper A100 and on higher difficulties. Also add a unique level feel message for these new Nightmare-enemy only levels and for spawning in Nightmare Demon/Cacodemon/Arachnotron/Elemental caves.
*From the Badges post of the OP, implement the suggested changes to the listed Bronze, Gold, Platinum, and Diamond badges, as well as change the Heroic badge series to be percentage-based as suggested (10%/30%/50%/70%/90% of medals for the Bronze, Silver, Gold, Platinum, and Diamond tiers respectively) to account for the new medals added in 0.9.9.8 and possible addition of any new medals in this or future versions, and add Heroic Angelic for obtaining all medals.
*Also from the aforementioned post, implement the suggest changes to the Shotgunnery Cross, Marksmanship Cross, Malicious Knives Cross, Sunrise Iron Fist, and Gargulec Medal.
*A toggle in the settings to disable the overlay filter of powerups.
*Some more of the proposed new exotics, uniques, and assemblies listed in the New Content post of the OP (don't need to do all of them now, but some more new toys than just Grappling Armor and Blood Boots would be sweet).
*Finish touching up the Apostle, in both its A100/666 fight and Hell Fortress fight, by increasing its teleporting chance per turn and implementing the suggestions for the Hell Fortress fight in this post (CTRL + F "Spruce up Hell Fortress' Apostle fight" to quickly find the exact suggested changes).
*Add method for guaranteed Dragonslayer and Berserker Armor in A100/666, so full wins in those modes do not remain terribly luck based. (If something more elaborate cannot be thought up, current idea is a 100% killrate up to floor 50/333 will spawn a Berserker Armor if you haven't found one already, and a 100% killrate up to floor 99/665 will spawn the Dragonslayer if it hasn't already spawned, while possibly locking the guaranteed spawns behind UV/N!).
*Allow access to Hell's Armory, Deimos Lab, The Vaults, and House Of Pain on ITYTD (I can think up of enemy loadouts for them if needed, just Deimos Lab absolutely needs to have only one Shambler on ITYTD so that it won't be a death trap for newbs).
*Add the new special level pair on floor 4, currently conceptualized as Nuclear Plant and Toxin Refinery.
MID PRIORITY
These are things I would like to see still, but they are definitely expendable to postpone if need be:
*When Enraged Mode triggers, have the screen shake and have all remaining enemies' act sound play at once, for more aesthetic flair.
*The music can be turned off if too many sounds play at once (this only really happened to me when I spammed run -> wait near a lot of enemies, which given the Enraged Timer now invalidates, this isn't too important to fix if it's not simple).
*When picking up a Unique for the first time in a run, the accompanying green flash may not happen, though it's unclear if this bug is with certain Uniques (confirmed it happened with Enviroboots and Medical Powerarmor) or happens for other reasons. The yellow flash when repairing armor/boots with an Armor Shard also sometimes doesn't happen for unclear reasons. Both of these bugs are purely graphical however, as the Uniques without the green flash still get counted as collected, and your armor/boots are still repaired by Armor Shards regardless of the yellow flash not happening. All other flash effects appear to be working as they should.
*If you hold the wait command for an extended amount of time while enemies are revealed, enemies' sprites cannot visually keep up with their actual movement. As a result, when you let go of the wait button, the enemies' sprites will continue to move around until they "catch up" to their actual positions. This bug appears to be purely graphical however, as proceeding to do other things before the sprites "catch up" will have the game work correctly off of the enemies' actual positionings.
*Various minor text/UI errors listed in the OP.
*Add default sounds for all exotics, uniques, and artifacts.
*The rest of the suggested equipment changes in the Equipment section of the OP that weren't specified as High Priority prior.
*Add sounds for enemies spawning, reviving, and Archviles charging their attack.
*Add corpses for Elite Formers and Lava Elemental, as well as a corpse sprite for the player when they die. The Angel Of Death also currently lacks a corpse, though maybe that could be left intentional as a mercy in case it's added to the A100/666 spawn pool, or in case the player kills it in Unholy Cathedral before killing all the other enemies on N! to trigger the Cathedral collapsing, thus ensuring it can't respawn and force the player to kill it again.
*Allow the Angel Of Death to spawn in very deep floors of Angel Of 100/666 (suggested depth is 90+).
*In House Of Pain, seal away the Large Health Globes behind walls or doors that can only be opened with the Arena Master's Staff, farther incentivizing to carry the Staff along for this level.
*Disallow nukes from being activated in Hell Fortress, and disable any active powerups when the player enters Hell Fortress, so that they cannot trivialize Carmack with a Floor 23 Invuln or Berserk stack.
*Have dying to John Carmack count as a normal death instead of as a partial win, so that there is more risk in going for a full win.
*Expand the amount of possible Dual Angel combinations, as covered in this post.
*Allow the action command to open armor and ammo crates. Alternatively, allow them to be opened by bumping into them.
*Add more Angelic Badges, so that there is an equal amount of them compared to the amount of badges in the other badge tiers. Go to the Badges post of the OP for ideas.
*Any of the other new medal and badge ideas in the aforementioned post can also be considered mid priority.
*Adjust the higher level rankings to require more badges of the lower tiers, instead of capping out at 15 for each badge tier. Also make No-Life King require every badge, not just every Angelic.
*Create/readd locked vaults as they're described on the wiki, as well as add the proposed new Trap Rooms in the Features post of the OP. (It would be very cool to have these, but I don't think they'll be easy to add and test).
*Any of the new level events proposed in the New Content post of the OP. (Though getting at least the Max Carnage and Snake Eyes events would be neat)
*The new Chained Court alternative level, that is currently conceptualized as Central Processing (though Kantarza may already have that in the works?).
*When you have the Berserker Set equipped, the infinitely spawning Nightmare Demons and Nightmare Cacodemons severely complicate clearing special levels, so some caveat should be added to make clearing special levels with that set a much less tedious ordeal (whether not counting the spawned Nightmare enemies towards a special level's completion, disabling their spawning in special levels, or giving them a spawning limit of some sort so that the Berserker player will be guaranteed able to get everything dead eventually).
Low Priority
These are things I would still support being done, but yeah you don't need to worry about them right now unless they would be very low effort to implement:
*Ensure bridges that go over rivers aren't blocked off by walls or rooms with no doors.
*Give EXP for pulling Levers, but consider disabling it if you have Intuition.
*Add color coding to badges and medals in the endgame mortem.
*Disallow Dual-Angel challenges for any badge that requires a specific challenge. This includes Berserker Bronze Badge to Pacifist Bronze Badge, Berserker Silver Badge to Pacifist Silver Badge, Berserker Gold Badge to Arena Gold Badge, Berserker Platinum Badge to Everyman Platinum Badge, and Reaper Diamond Badge to Arena Diamond Badge. (Aside from effort concerns, I'm also putting this in low priority because I'm not so sure I still support it, Sereging is very cheesy but maybe players should be allowed the extra creativity when badge hunting that Dual Angels allow. Maybe though just address some of the most egregious cheese that clearly goes against the spirit of the badges, like the Destroyer badges not actually requiring their respective Untouchable medals and all the badges that require Explorer/Conqueror not actually requiring those medals, thus resulting in those badges being much more easily won in AoOC despite it disqualifying the earning of said should-be requisite medals).
*Allow previously obtained badges to appear in mortems, and keep count of them. To prevent the awards section of UV/N! mortems from becoming absurdly large, have only the highest tier of each badge appear on the mortem.
*Have more badges require at least a standard win, rather than any win. Current suggestion is every Gold, Platinum, and Diamond badge that requires winning the game should require at least a standard win, with the exceptions of those that already require a full win and Pacifist Gold.
*Have the trait selection screen be more color-coded (see the Features post of the OP for the example).
*Add a time display to the Player Info screen.
*Any of the new challenges proposed in the New Content post of the OP.