Mains Linear Power Supply Protection
Is a fuse really going to help prevent damage to the circuit?  


In general, a fuse isn't intended to protect the circuit. The fuse is there to disconnect your circuit from the mains in the event of an over current. That way, the circuit breaker at your facility (home or otherwise) doesn't shut off and lose power to other branch circuits in your facility.  Also, your over current might not be enough to trip the facility circuit breaker, yet enough over current to start a fire.

Other devices such as TVS (rather to use MOV) (after the fuse) ARE desired to protect your circuit from transient (temporary) peak voltages. A transient over voltage can be clipped without blowing the fuse. Fuses take a long time (relatively speaking, look at fuse specifications) before they open.  


  Should I also fuse the mains neutral  


Oh, no, never. Disconnecting the neutral is not a good idea. This leaves the HOT lead still connected, and there are many questions and answers on this EE SE site regarding disconnect of neutral.  Like this answer