Gintama.
Just watch it.
And few more none mentioned yet:
-Darker than Black; it goes under "superpowers" but I think its great
-To Aru Majutsu no Index, once again there are "abilities/powers/whatever you call them" but its pretty awesome. Ends kinda lame though, but I think its going to get a sequel spin-off kinda thing (think tales of symphonia style sequel)
They don't have those 3 points you dislike though. Maybe for second one, second "point" applies, but not because hes strong or something like that.
Oh and watch Princess Mononoke. Its beyond amazing.







