I wonder if I can mail a Meele weapon from a Klingon character to a Federation character.
Yes you can, as long as you didnt bound the item to one char, and as long as the item is useable by both sides (most item does, it seems that only BOs are useable from their original faction), you can check it in the item tooltip.
So far i didnt tested it for melee weaps, but if the item is useable by both sides, then its ok.
Unarmed combat DOES have a combo system builtin with various special effects and differences from the Bat'leth.
Likewise you DO get tactical kits and other abilities that make your melee better, your unarmed melee. This is as intended and is a part of factional flavour, devs even spoke about it in an interview or two if you care to dig for it.
And yes you can mail over whites, the greens from the medal vendor are bind on pickup, so no those cannot be mailed.