Java/Arduino Controlled 120VAC Power Outlet
That sounds like your Arduino is getting corrupted from noise spikes on switching the AC load on and off.  The reason a glue gun affects it less, is that as a pure resistive load, it doesn't generate very much noise, but the alarm clock probably has a switching power supply in it that is dumping higher frequency noise back on the line. 
Even an older motor driven clock can put spikes back on the line depending on the motor and how it is controlled.  Have you tried using an AC filter between the relay(s) and the load (Not a surge suppressor, but an actual LC filter)?

It is also possible, though from the description, unlikely that the relay coils themselves are generating enough of a spike to corrupt your processor.  Industrially, for just that reason, we use RC snubber networks across the coils of just about every relay/solenoid driven by an electronic output.

Also the above advice is 100% correct, do NOT switch the neutral of your AC load.  Very bad business there from a safety (and at least US Code) standpoint.  If the neutral must be switched, to be code compliant, you would need to use special safety relays with force-guided contacts in series.