For most of these words the key thing to pay attention to is the root word which indicates the item involved. Untie is only for things you tie together, like string or rope. Unbuckle involves buckles (like on belts). Unhitch involves hitches, like you have on the back of trucks for trailers. Unfasten is generally for certain things that you tighten for safety (fasten is a verb for tightening ropes and such, or making sure it "holds fast"), like seat belts, maybe climbing ropes... Unbind isn't used that much, but is very like untie. "Binding" things together sounds less deliberate and more messy, as it can happen by accident. Something in a mess of tangled wire could be said to be "bound up." It can also be used in talk about molecular chemistry, atoms are "bound together" in molecules. For me, it makes me think of magic spells in fairy tales. "Binding spells" that trap things together or in a certain area. Undo can be used in reference to most all actions. Anything that can be done can be undone. "Un-" reverses or negates the action in question. If you tie something and want it not to be tied anymore you "untie" it.
Hope that makes sense, phew, it's actually surprisingly hard to talk about without using the words or prefixes themselves!