How to connect 120V to a relay on a prototype board?
Yes, you can. Research "creepage and clearance" to be sure you route the high voltage conductors properly.
Yes, you don't need a third wire ground.  If you are mounting this whole thing in a metal chassis a third wire ground would be a good idea.  (Connect the ground wire directly to the metal chassis.)
Depends on your maximum load.  Search for AWG vs. load to see the appropriate recommended conductor for your lamp load.
Yes, assuming you follow the creepage and clearance and insulation isolation rules, as well as sizing the traces for the amount of current.


A fuse mounted somewhere in series with your AC line input would be a good idea as well.