Hi
Is possible to get power from the stereo like you said. well, actually it's pretty much the same if u using the 12v cigarette socket or the stereo hardness.
next is:-
there's a battery pack in the sat nav. As i've cut out the battery pack for safety issue (battery looks kinda unreliable )
yes, the unit will stop working when the ignition is off.
unless the battery pack stay intact .. you could still use it until the battery dead and recharged when the ignition is on.
luckily the unit has it own save memory setting and so u won't lost all the setting although it only get power when the ignition on and without the battery pack. Not sure about tomtom as i never own one.