Fact for you Trivial Pursuit buffs: Glass is a liquid.
Everything you said was right up to this point. It's been repeated often, but glass isn't really a liquid.
Back to the ice, the level of the sea changes when the piece of ice falls in, not when it melts. Ice is less dense than liquid water, so a given volume of water produces slightly more ice (which sticks up above the surface when the ice is floating). When the ice melts it returns to the density of liquid water so the surface stays exactly where it is. Evaporation doesn't play any part in it.
Water is unusual in this behaviour. For most materials the solid does tend to be denser than the liquid (and much denser than the gas). As you cool water it does become denser, but only until you reach about 4 degrees Celsius. As it cools below that temperature (and eventually freezes) it expands. That's why frozen pipes burst.