1.) You're supposed to, but I still have some bicycles registered at my previous residence in Tokyo (I'm in Kanagawa now). I have never been checked in fifteen years. In my experience, MTBs and road bikes are rarely ever stopped.
2.) Yes, it is. You're actually supposed to ride on the road, though some cops will claim the opposite. Some pavements have dedicated bicycle lanes.
Most motorways are fine but you will see traffic signs such as the one below on roads which are off-limits (as are fly-overs, many bridges, and, of course, highways).
3.) Just anecdotal, but I have never seen anyone getting busted for drink-riding. If you do cause an accident in a state of intoxication, however, you will be in trouble.
Generally speaking, I wish the police would enforce traffic regulations more strictly on cyclists (and not only go after stolen mama-chari or pretend to pay attention during traffic safety week).
4.) Sign up to our sister site. We have a classifieds section. Also, our members will be glad to help you with all your questions.
Tokyo Cycling Club
tokyocycle.com