Do they utilize both skill sets of Conjuration AND One-hand/two-hand/bow weapons, or just one or the other?

Bit of both. The Conjuration perk "Mystic Binding" conjures stronger weapons, but you're gonna need to put points in One-Handed to use Bound Swords effectively. Same for Two-Handed and Bound Battleaxes, and Archery for Bound Bows.

Also, Conjuration perks past Mystic Binding also offer a perk that casts soul trap when you strike with a bound/mystic weapon, and another that banishes daedra.

Great info! Thanks very much Galla!

You're very welcome!