I've had this discussion many times w.r.t. global warming, and am amazed so many people, including scientists, just don't get it. The level the seas would rise is not so much dependent on floating ice (north pole) since the displacement is exactly the same, but on ice that was on top of land (e.g. Greenland, Antarctica, etc.).
Absolutely, although it's worth noting that loss of floating polar ice will change the albedo of the earth, so there's a positive feedback loop there that may result in further warming & consequent melting of land-based ice elsewhere.
So melting of the floating polar ice will not directly result in raised sea levels, but 2nd round effects of the melt might do so.
The ice cube was previously floating and was therefore displacing 10g of water.
If we let our ice cube melt, we will have 10g of water than we can add back to the glass.
The net result is that the water level remains the same.