Well we know gravity manipulation has been mastered by Archers time, so it's not out of the realm of possbility (in my mind anyway) that the saucer (spear, spade?) of Voyager would have some sort of anti grav plating or mesh in it to lighten the load so to speak. Or maybe they used a light tractor beam to anchor the back of the ship to the ground.
Well we don't know if that (theoretical) system lost power, or if they just shut down the systems they were doing work on. There are more power sources on that ship than just the warp core.
There are more power systems on Starships than the Warp Cores. There is the proverbial Auxiliary Power. The nature of which isn't precisely stated, but it's presumably it's a number of fusion reactors which can use the deuterium. There is Emergency Power mentioned a lot which may or may not be a true power system. It may be just power that is held in reserve to be used upon Captain's order. And then there are Batteries which apparently can propel a Starship at sublight speeds and maintain orbit for short periods of time.