Can you do passive enchantments on weapons? What I mean is, can I enchant my weapon with something that I won't have to recharge ever, like a fortify stamina or something similar?

No.

I'm 99% sure there will be a mod for this once the CK is released. I will be installing it, for sure.

Haven't been able to since Morrowind.

I know. I missed that in Oblivion because I can't be bothered with recharging weapons all the time. The only enchanted weapon I ever used in OB was dawn/dusk blade because it recharged itself every night and day. Ah well, no enchanting the weapons this game either I guess. Either way, Skyrim still rules.

Using an enchanted weapon is the easiest way to level enchanting, you get skill ups from recharging weapons, but I don't see why you do