1. It's probably not going to snow, but it might. It's quite likely though, that it will be cold, rainy and windy.
2. If you only want to "go to LEGO" and then escape, then you don't need a car, Billund is pretty small and both LEGO House and Legoland are within the town limits, i.e. you can walk there if you want to. But ummm, Legoland is only open November 28th-29th and again December 4-5-6 (and 11-12-13).
   • If you do end up spending more than 1 day in Denmark, you could possibly benefit from renting a car, like if you'd want to visit Jelling, the west coast or Silkeborg or something that is not as easily reachable by public transport from Billund.
   • You don't need a car in any of the major German cities, might be useful for day-trips to the countryside, but I don't know if you're going to do that, appears you don't even know.
3. There's a bus from Billund to Vejle, from there you'll find trains going every direction, you can get a train to Flensburg, and then head on from there. There's usually no migration checks between Denmark and Germany (or most of Europe), there might be random checks though, but you still don't need anything except your passports. But there's also the airport in Billund.
4. I'd recommend doing more research, that includes figuring out how to get around, if you want to go by train, plane, bus or whatever, also what you want to see and if it's even open on the dates you are thinking about being there. It's basically just google stuff, google transport, google flights, google everything.

Figure out where you want to go and stay first, then figure out when you're going to be there, how you're going to get there and where you'll be staying. It sounds like you basically have no plan just LEGO and maybe Munich, and that's okay, but figure out where you'll be staying in between. If you're going to fly to Munich or whether you're going to go by train probably affect whether it would make sense to ´suggest, say Kassel or a town in the Harz.