Two questions. 1. An ice cube is submerged in a glass of water. 2. An ice cube is floating in a glass of water. When the ice melts, does the water level rise, fall or stay the same in either case?

in the first case.. The ice cube is less dense than water. As it melts the water volume occupied is less. The water level will fall a small amount.

the second case
the level will stay the same.
The mass of the ice cube = mass of the water displaced. The volume and density difference is the small amount of ice above the surface.. Once melted the mass of the system stays the same. the ice gains density and looses volume to equal the volume of the water it displaces.
(if you look really close the system will shrink (water level reducing) just a tad when going from 0 to 4 C..
and then start to expand again. )
Hope this helps