Is there a shuttle from airport to hotel?

Yes, but there 3 airports in LA. Check with the hotel about service. Burbank (Bob Hope) is closest. Remember, the hotel is part of Universal Studios.

No they don't. The closest airport is Burbank Bob Hope and it costs 10-15$ with uber and 20-25$ with a taxi.

Yes

I just stayed last week and no there is not. Uber will cost about 25 dollars and a taxi will cost you 43. I had called the hotel at 3 different times to ask, so you get different people. They all suggested Uber.

Yes. Most Major Hotels provide that service. Just check in with the hotel.

Yes, assuming you fly in to one of the airports they service. Worth checking out before you book if possible. I would pick Burbank if I had the choice!