Here's a quick breakdown of my impressions of the various options available to you.
Warrior: Complex and rewarding tank spec, simplistic me hulk me smash style DPS specs. Best overall movement of any class in the game, quite a fun aspect if you like zipping all over the place.
DK: Simplistic DPS and Tank specs, have moderate and rewarding levels of finesse if played very well, both very powerful in ther respective roles. As befits soloing old raids, you'll be hard pressed to find a better class. (Blood damage taken and damage done is unparalleled.)
Monk: No personal experience here but it's looking very strong in all specs.
Paladin: Simplistic and powerful heals with a very nice toolbox, the use of which is your finesse. DPS spec is boring IMO, and dependent on execution of burst phases with periods of low damage in between. Tank spec is more complex, very rewarding to play well.
Warlock: Doesn't meet your criteria since it can only DPS, but as I am a warlock fanatic I'm going to say a few words anyway. You will be hard pressed to find another class that offers the depth of complexity and flexibility of playstyle. This class can be applied to content in creative and rewarding ways that just aren't available to the simpler classes, even those with multiple specs. Stygian warriors with one foot on the other side. Shadow, fire and pestilence are your tools, death is your trade. You are the harbinger of the apocalypse, bane of the living. Plus you have a purple raid frame and look badass in black.
I think you'll have fun no matter what you choose though, 4 expansions and 8 years, blizz has gotten pretty good at refining the various classes.