Re: crossing borders on bike, I wouldn't expect any particular difficulties within continental EU as long as you're taking a road where biking is legal in the first place. You would obviously have to make some arrangements if you plan on e.g. crossing the Channel.
Possible restrictions in those situations would be related to whether or not ferries / trains take bikes and at what costs. For example, there's a freight ferry connection from Finland to Germany that also carries passengers, but they specifically state passengers must have cars ("because the only way to move through the restricted port area at either end is by your own car"). When I asked about it, turns out bikes would be OK as well.
Outside of EU, different story, but even then I suspect it's more of a Schengen / visa issue (e.g. EU to Kaliningrad or vice versa) than a question of your chosen means of transportation.