I didn't know you could use an arduino without providing exactly 5 volts. If I use rechargeables AAs, 1.2 * 4 is 4.8 volts. Would that damage the arduino at all?
No, this is just fine for the AVR and the FTDI chip (3.3 to 5.3 is a safe margin). Note that fresh 1.5 batteries have an open circuit voltage of around 1.65 volts.
Also, can you program an RTC to give a pulse every X seconds?
One second will be fine as well, just go to sleep (x-1) times again; my alarm clock calls that "snooze"
As for a switching supply, I assume that that is a power supply that turns on or off when it detects a pulse? In that case, the RTC would give it a pulse at t = 0, the arduino turns on, then maybe the RTC (or arduino, I guess) sends another pulse at t = .1 seconds or so to turn the arduino off. The cycle then repeats?
Quite similar. You will need an external chip, a "retrigerable one-shot". A switching regulator (as the popular LM2574 or LM2575) needs a permanent "On"-signal. They generally works between (Voutput+1) and 40 volts. So it is fine to use 12 or 24 car batteries for suply. There are also so called "boost" regulators which convert from 3 volts (=2x AA cells) to 5 volts