Question 2 is the direct corollary of the Archimedes principle. When the solid ice is floating happily on the water, the volume of the water it displaces is the same as its weight (which is conveniently the same value as its volume using our SI units). Later when the solid ice melts, it turns into water which has the same volume as its weight, and this new water replaces the "hole" where the original submerged ice was. Since the original displaced volume is the same as its weight (= Archimedes principle) which in turn is the volume, the water level will not change.
Or to make it more concrete:
Let's assume that in the beginning, x ml of ice is submerged in the water and y ml of ice is above the surface. How much is the weight of the ice? It's x grams (due to Archimedes' principle). When it melts later, there's a "hole" where that x ml of ice used to be. How much water will we get from the molten ice? It's obviously x ml (as the ice's weight is x grams). So you have a "hole" of x ml but it's replaced by x ml of freshly molten water, therefore you get an unchanged water level.
I have not considered the temperature change that Frank has pointed out with much insight.