1. I think the lightest hard case you can find will be a better choice than a soft one. It will protect your banjo better and be easier to tie down securely. You could even mount some d-rings on the case to lash it down with. The hard case will also facilitate shipping the banjo home if you decide to abandon the idea.
2. Lashing it on the back with the neck hanging out is the only reasonable way that I can see.
3. Best solution is a new bike! Surly Big Dummy. Expensive solution, but the best for carrying odd loads.