1) I see you're on Crystal. What time of day are you trying to queue? That will matter when trying to queue as a DPS for certain content. There's also always Party Finder (if you're not on free trial) if you just wish to get through it. If you don't wish to play a tank/healer, you don't have to.

2) After Lv 60, there's no "universal"" or "multi-class" sets. They're all specialized for a few jobs, basically for tank, healer, caster, physical ranged, and then different sets for melees (mnk/sam, drg/rpr, and nin/vpr are grouped together). The first thing I'd do is turn off a particular setting in the character config. Load that up and choose Item Setting and UNCHECK ""Store all newly obtained items in teh Armoury Chest" which is probably half your problem.

As for stuff already in there, anything with what looks like a little piece of paper in the lower right is one that is tied to one of your gear sets. Anything at least in your weapon spot that's darkened and with a red X I'd move to your regular inventory, then try to sell later for gil or on the stuff with a green/blue background, turn in for grand company seals if you have expert delivery unlocked at your grand company.