1) PSP's are region-free so, yes, you can play any UMD game in any PSP that has a working UMD drive
2) Recommendations off the top of my head:
if you like SRPG's and Metal Gear, then get Metal Gear Ac!d 1 and 2
Disgaea 1 and 2
Ridge Racer is a great arcade racer
Gran Turismo is good in you're more into racing sims
Daxter is a pretty good counterpart to the Jak series on PS2 (I hear that "Lost Frontier" game is pretty good too, but I haven't played it yet)
Lumines is great puzzle fun
if you're willing to import, any of the DJ Max Portable series are fantastic rhythm games (Oh yeah, DJ Max Portable Fever was released here and it's basically a "greatest hits" version of DMP 1 and 2)
God of War: Chains of Olympus
GTA: Chinatown Wars
Prinny: Can I Really Be the Hero for hard side-scroller platforming
Patapon is an odd sidescroller/tactical/rhythm game hybrid
I may add more later