Generally, for scenario 1, call the aircraft in question, request his position and the nature of the emergency. Relay that info to whatever ATC/FSS unit you're working. They will then attempt to work the aircraft themselves. If this proves impossible they will have you try to switch him to another ATC/FSS unit closer to his position. If that fails, they may switch both of you to a discrete frequency and have you relay. That is the general rule of thumb. No two situations are alike.
For situation 2, check the time. If it's between the hour and five minutes past, it could be someone checking the operation of his ELT. That's only normally a couple of sweeps, tho. Listen out for a couple of minutes to discern if the signal is increasing or decreasing. Call the ATC/FSS unit you're working, tell them you've got an ELT signal and advise them if it's increasing or decreasing, and how long you've had it. They'll tell you what to do next.