I live in a 1 bedroom apt, and do stove top partial mash partial boils. My kitchen is kind of weird and I don't really have enough room to setup a chiller let alone store it.
Tho temperatures of water are never a problem. Even in the hottest summers public water in Canada is generally down to about 4-10C. I just fill the bath tub, and put the brew pot in there. Usually takes about 20 mins. I use that time to clean everything (save for the brewpot which soaks overnight) tidy up the kitchen, put everything away, and of course have another beer.
To me making ice was never even thought of. Not only am I weirded out by city water sometimes, but making ice? I live in a 1brd apt as stated. That means one ice cube tray, and little freezer room. To me the 20-30mins it takes to cool my wort, is a good trade off for having to boil, then freeze several batches of ice cubes.