Does damage immunity work on Urgot ult?

Yes for all of them except Morgana's (not sure on that one).

Morgana shield will work only if Morgana shields the target before being hooked to death.

Yes, in most cases advantage goes to the defender, though there are a few devs who want to make it into "true death" as in "haha none of that stuff works, you're screwed buddy!" along with even making it disable passives that only proc on death so if you're Sion, Karthus, or Kogmaw and get killed by that ult then "Nope, you don't get your passive because fuck you!"

But they're not doing that. For now at least