If you're on 6.60 or use POPSLoader for the latest version of POPS, the compatibility is near perfect. That said, there are a few games that run slightly better on older versions of POPS. For most people, using the the latest POPS is fine.
As for best PSP, that depends. Brite/Go, you'll get a nicer screen (more vivid colors, less ghosting), assuming the very minor scanlines don't bother you. Slim (for the most part) will give you good compatibility, full CFW, and proper NAND access. Phat has the best compatibility with older homebrew and CFW, but also only has half the RAM of the newer models. Most people will be fine with the Brite or Go.
If you desperately want multiplayer for your PS1 games on the PSP, then you can look into IRShell, but the multiplayer is buggy. It will never be improved, and will only work on Slims (maybe Phats too, I don't remember). An option I recommend the general user to avoid.
As for CFW, ME is the most up to date version, as it still sees some activity once in a while. It supports everything Pro does and more.