Yes, custom matchups and leagues are available on all the major sites/apps. This is one of the coolest things about playing DFS. You can play with your friends whenever you want.
NFL
There is a wide variety of platforms, but FanDuel, DraftKings, and Yahoo Sports are three of the best. Start by opening an account here and then you can venture off into some of the smaller markets.
Yes, every major DFS provider has a mobile app. This is a way they can prove the location of your device.
It is legal in all but six states across America. You may want to double-check with your state-wide legislation just to make sure. The main sites will let you know.