Would it be too weak? Could it be braced somehow to increase the strength?
Yes. Maybe. Sure. Depends on the design.
> 2) What is the best way to join MDF? Dowels? Biscuits? What is the best
> type of glue to use?
I use biscuits and any good wood glue.
> 3) What is the best way to finish the edges of the MDF? Should it be edge
> banded with hardwood?
Not if you are going to paint it. .Edges can be routed if you want a shape or roundover.
> 4) I plan to paint all the MDF. What type of paint should be used with MDF?
I have little experience there. Prime it first though. I used spray paint, but any should do.