>>1398145I can't speak to hitchhiking, but I've biked all over the country and I've enjoyed it quite a lot.
Without very specific route planning and/or adjustments, a hammock is not the right tool. There are many big parts of the country with no trees.
As far as routes go, I've crossed the country twice from north to south, once from east to west and once from west to east (all by bicycle, camping every night). Going north to south (or vice versa) lets you stay in/near mountains the whole time, like the AT, PCT and CDT. If you care to cross from coast to coast, I highly recommend staying north, at least east of the Rockies. VT, NH, upstate NY, UP Michigan and Minnesota are all lovely, and the national parks around MN, ND and SD are very underappreciated (Isle Royale is amazing).
When you want to leave will depend on your route and your pace. 3 months is generally a good guess for biking from coast to coast.
