Considering MAc means there are no defensive traits, have I been doing the right thing by Power and Bulk modding Red Armor, considering the trait is really more oriented towards killing everything before it can kill you?
Ammchain is an offensive trait, yes, although the implications of infinite ammo certainly give you an advatage for defensive purposes, specifically what tends to be called "tactical defense". Since you have as much ammo as you could want, you're also allowed to waste as much ammo as you want in the form of corner-shooting and radar-shooting. This means in situations where you can't see the enemy but can theoretically hit it, you're better off hiding in a position that lets you shoot in their direction. Naturally speed won't matter in these situations.
Is it worth it not being able to dodge worth a penny?
If you really need the ability to dodge, technically you can take the armor off. The 15% dodge chance and 15% movespeed bonus on Hellrunner are actually the same thing (at least according to some people with source privileges), which implies that movespeed and dodge chance are directly related. It really depends on what you're trying to dodge, though: naturally VMR packs won't do you much good, but Barons/HKs/Cacos will. I'd probably keep the armor on for Arachs and Commandos, even though it burns though your armor, just because they hurt that much more without any.
And, since I've never used Cateye, is Cateye really something considered an awesome trait that shouldn't be passed up?
I'm a fan of Cateye. It flows naturally from key rapid-fire traits, SoB and EE, and that extra vision can really makes things easy on non-segmented floors. Ammochain is great because you can then afford to carry lots of extra gear, but you're also forcing some agility mods on your weapons, an entirely unnecessary part of Cateye builds (and pretty significant when you think about what else can be done with those agility mods). The difficulty lies in having ammo, but that's why it's a good idea to keep a chaingun and dual-stock on bullets and plasma. As superior a weapon as the plasma rifle is, a properly-modded or assembled chaingun can be more than sufficient.
Is MAD a good trait to have, or should I ditch it for something involving Whizkid and Cateye? Would, say, getting in their faces with a Double Shotgun be viable, or is Cateye really the way to go in this particular case?
Army of the Dead is a passive offensive master, basically giving you extra damage on your attacks. To this extent, you can't really play a very open shotgun game, relying mostly on the combat shotgun and corner-shooting (although close corners let you whip out the double). Power-modding a shotgun, however, can be just as powerful, and some of the shotgun assemblies are ferocious. It really depends on the type of game you're facing: if you want high kills AND on a hard difficulty, MAD is probably for you. For anything else, SM/WK or MFa (or both) are plenty reasonable. (Mind you, Cateye as a shotgun build is mostly useless: shotguns are completely unaffected by accuracy, so the extra vision will do nothing for your damage.)
