Yes I do. As I see it, not naming a pokemon on your party is like keeping a pet and just calling it "Dog". Because the given name is the name of the species. Also, it adds a bit of personallity to them. I won't use a pokemon on my team if I can't name it myself, which is in sharp contrast to most people.

I make an exception to legendaries though, because they're mostly considered to be the only one in existence. (in the story) For example, I kept my shiny Giratina's name but I named my Shiny Magby "Maggie".

I dunno, I have way more then six pokemon. But some of my personal favorites are: my Mamoswine, Bootstrap. He's gotten me through countless battles despite his below average speed. He also comes in handy against the common scarf chomp. Also my Ambipom, Bramble. Bramble beats the snot out of any sweeper at or below 115 base speed. I breed him to have near perfect speed, attack, hp and defense. My two walls, L'etoile the Lanturn and Pierrot the Tangrowth are also very awesome. =]