delay does not cease code running - processor is still cranking away at 16 MHz, it's just not doing anything but counting the time going by.
Power down sleep mode vs sitting in delay will definitely save on power.
I don't know about the durations. I made an RF remote control using a promini powered from LiPo battery. Pressing a button created an external interrupt to wake it up, read the keypad & send the button press, then go back into power down sleep mode. Runs a month or so on 1000mAH LiPo, just sitting there with no button presses. I should rebuild it into a standalone module with no power LED, no regulator, and p-channel mosfet to ensure the transmitter is all the way off.