You have to build the pirate ship before you get the new island quests. You do NOT have to have UFO open to get the quests for the island pieces. There is a quest to get a piece of the island that needs the UFO open, but you can get started on the whole string without it. It says you can buy the island pieces, so I suppose you could pay 25 diamonds and get island without UFO.
I'm still waiting on GI to let me know if banishing robots will count towards my robot quest since the new update. It hasn't counted the last 2 updates, so I haven't been able to do anything on UFO quest for weeks!