Should I monitor the emergency frequency (121.5 Mhz) at all times?
Airlines routinely set comm 2 to 121.5 when they don't need it for talking to company or getting ATIS.  But for a light aircraft pilot in busy airspace, the second comm is a really useful tool for reducing workload by being able to preset frequencies, and dedicating the second comm to 121.5 all the time takes that feature away, for the very small chance that you're hear someone's ELT go off or broadcast an emergency that will be heard by ATC anyway.  Very few GA pilots monitor guard frequency regularly.

On the other hand, if you are flying over a remote area, it's not a bad idea at all and you could save someone that crashed with a non-406 ELT.