Welcome to the forums! I hope you enjoy your stay. Remember, the only stupid question is one not asked.
The Halls of Carnage used to be much easier - I don't know why this was changed *grumble grumble*. You pretty much hit the nail on the head though - you need to blaze through at top speed. The lava columns start coming from the left at 1 column/3 seconds on ultra violence, so you need to move quickly.
Firstly, before I say anything - unless you're going for a badge/medal that requires you to be conqueror, take a look at how you're kitted up before deciding to enter a special level - No BFG is worth a dead doomguy, because you can quite easily complete the game without one - you can't complete the game if you're dead.
Secondly, this special level is paired with the Spider's Lair - meaning that there's a 50% chance of them spawning interchangably on the same level. If you ARE going for conqueror and have heaps of trouble with one special level, it may be worth simply biding your time until you get the level you can complete.
Now, as for the UV HoC:
-Move quickly. And by this, I mean move quickly quite literally - you'll want your movespeed to be as high as possible, since the lava doesn't move based on turncount, it moves based on time spent on the level. The Red Armor in your example would have given you -20% movespeed, for example. Keep in mind that shooting and reloading weapons takes time - 2 shots and a reload from a shotgun would mean one more column of lava to contend with.
-Plan your route. There's 3 full heals in this level - keep in mind that each of them resets your running status to cautious. If you plan it out, you can literally run through most portions of this level.
-Abuse Sergeants. Their shotguns can be annoying as hell, but if you notice an opportunity where a Sergeant is to your left and you're not low on health, stepping one step closer might take you farther via knockback then if you had stepped to the right. Mind you, if you're doing a damageless game you can't do this, and if you're not careful this will flat out kill you.
-Lava hurts. Not just you, but it also hurts monsters. If you're going for 100% kills, all you really need to worry about is the lost souls in the middle section of the level. If you're not - great! Just run past them. Your objective is to get the BFG and get out alive, not to be a hero. The stairs to the next level are never affected by floor statuses - you'll never have a corpse/blood/lava/acid/whathaveyou on stairs, so once you're at the stairs you can simply runwait for some free exp or something. Keep in mind the walls are also destroyed - don't be too cocky about this or you'll end up getting 5 acid balls in the face from 5 'almost dead' Barons.
Hope i've helped!