Again, I hope I don't mess up any details. You don't know that much after one time.
1. Who defeats Ultron at Avengers' Tower when he first appears?
Clint throws Steve his shield - just one example for his perfect characterizing in this movie since he's one of the fewwho can handle it in the comics too - and Steve throws it at Ultron. I think that's the last straw for his first form to crumble.
2. Do we actually see Thor traveling to other realms during his solo quest for answers?
Nope, he's just having visions.
3. Were any of the Avengers shocked when it was revealed that Hawkeye had a family (wife and kids)?
All but Natasha.
4. Did anyone else, other than Thor, help in the destruction of the flying city? Would you say that the moment allowed Thor to shine in the same way that Iron Man carrying the missile through the portal (from the previous film) did?
It was team effort, from what I remember. The others all helped evacuate the city. Tony was there, I think, but not doing much to the actual blow up of the city, Wanda destroyed the core that had the city flying, from what I remember. But it's mostly Thor.