The only reason most hospitals will keep you if you're not in active labor is if your water has broken. Otherwise, 3 1/2 cm is not enough, usually not even 5cm. You can dilate like that for weeks before going into labor unfortunately and it's not really a sign that you'll go into labor early
Likely not if you're not in active labour...
Probably depends the specific hospitals policy. Some require 4cm some may not keep you even at 4 if your not having contractions. Only way to know would be to call labor and delivery and ask them. I personally wouldn't recommend going yet if you're not having regular contractions. It could be a very long time before you do and home us much more comfortable then a hospital. I would recommend getting some rest and preparing.