If I applied an untradable nametag onto a tradable weapon, does the weapon become untradable or is it still tradable?

No, it will not change the tradability of your item. If your item was originally untradable, your item will still be untradable.

I was a bit unsure of this as well, so when I opened my Smissmas Stocking, I tested out the Desc. Tag on a random weapon that was tradable. After using it, it had no effect on the tradability of the weapon itself.

Any tool item works with the exact same mechanic. For example, though Keys will be listed as "unable to be used in crafting", whatever you get if you open a crate can be crafted (as long as it's a weapon or cosmetic). The same will go for Name Tags and Backpack Expanders; they will not effect the tradability of your item/backpack.