Does it power off completely or go into a deep sleep state?
No matter what, if there is no battery, there is no power to do anything, it is dead. But android saves the state of these even if you pull the battery. If power is restored before the scheduled alarm, it will still fire. (Assuming you are talking about the actual alarm app. It is feasible, but unlikely, that a poorly written app's alarm scheduling could fail with a power failure.)
However, if you restore power after the alarm has passed, the alarm will not fire.
TLDR: It can't be in a low power state, if there is no power. But it saves scheduled alarms, etc, to memory, so they are not lost.