Is ice and ice cream the same word in German?
I would say no. Eis is more general. It is the word for frozen substance. And then we specify the ice. "Wassereis" "Speiseeis" "Eis der Antarktis" "Eis aus Kohlendioxid" and so on. "Eiscreme" (ice cream) is usually the ice made from milk/single cream for eating. In colloquial language of course you just say ice if the context is clear.
Depends. We refer to ice cream with both "eis" and "eiscreme", but in daily communication it's 99% "eis".
If you want to talk about frozen water in general (e.g. in a lake) only "eis" is correct.