Everything can be done solo except the optional raid bosses and maybe the optional arena battles. I found Moxxi's Underdome not fun to solo in BL 1 so I haven't gotten around to trying Fink's arena in 2.
Fink's and Ore Chasm are more like the other arenas from Borderlands 1 such as the Circle of Duty at the end of Knoxx. I've never tried Moxxi's since I always played solo, so I can't tell you how it compares to those.
EDIT:
I thought I'd add that I beat the Captain Scarlett DLC last night, and was a bit disappointed when I got screwed out of getting more loot. :| I saw the badass achievement for finding the hidden loot area, and I thought it was on this ledge that looked like it would be accessible from climbing up the other side of The Leviathan. Well, it turned out that his other flipper was not scalable, and I fell through it. Unfortunately, after that point, the gate to get back in was closed and I couldn't access any more of the loot.
I was going to toss loot over the edge so I could pick it back up and sell it at the vending machine in there.
None of it was really that good, and I took the most valuable stuff to begin with (since I wasn't sure if my idea was... solid :biggrin
, so it wasn't a total loss.
Although, I can't really find a weapon that I like using against Sand Worms. They're highly resistant to caustic weapons, so my super awesome E-Tech Caustic SMG doesn't really work against them and its damage is so ridiculously high for its level (level 30, 800+ damage). Oddly enough, I've been unable to find any other E-Tech weapon that has nearly that much damage on it. I found a Hyperion Incendiary E-Tech of the same level, and it was only about 350 damage, and had the usual Hyperion Michael J. Fox aiming system.