The best player made food and drink will speed it up, but even with vendor bought food and drink (or none at all) you still regen at a very nice clip. While out of combat you rarely 'med'.
Its nothing like EQ1, where I remember literally going afk for 15 minutes while I parked in a safe spot, so I could get back to full mana on my druid. I know they sped it up since then (this was a couple years back) but EQ2 is nothing like that.
Sitting has 0 effect. They were going to have it speed up regen slightly, but they took it out before it went live. Or so SOE has stated. I've tried sitting, and I believe them, it doesn't appear to have any effect at all.
Any class can solo, it depends more on the player than the class. If you are intelligent and can figure out what you are doing and what mobs to take on, you can solo. If not, it doesn't matter what class you are.
If you want to solo easily, pick a paladin or similiar class that can tank and heal themselves. I personally believe the paladin is one of the easiest classes to solo (then again, I play one, so I may be biased). And personally, I think they are a really fun class.
My newbie shaman solos really well too but is too young yet to make a judgement call.
I watched a guy the other day,a mage, and he soloed extremely well, although we couldn't always do the same mobs, and he had to have a lot more strategy than I do. I just wade in and start beating on the mob, he has to keep aggro on his pet.