Generally beams should rest on top of a column so that it is the only thing supporting its weight. If a beam comes into the side of a column, you're now relying on both the column and the join to withstand the load. Joining a bridge into the side of the ground doesn't really work. You can bury your foundation into the side of the cliff to more or less achieve the same effect. Just keep in mind the bridge needs to rest on top of something (or you need really strong joints).
Bridges generally don't need to extend past the gap that they span. Anything that goes beyond the furthermost supports is going to be acting in tension, and it isn't going to be doing much more work than the members already in the truss unless the material specifically acts much better in tension than the material doing the spanning (i.e. you might tie a truss down to the ground with cables). However, most trusses are going to be designed to resist everything via the spanning portion since its usually cheaper.