If the sink is shallow, try the bath tub.
Ice in the wort is bad (the sanitized bottle thing has been mentioned before). Basically if the ice isn't sanitized first, you are pitching back in bacteria. Additionally after it is boiled, it has to be covered until used - ie you can fill ice cube trays, well I guess you could if you put a piece of plastic wrap over them, but yes, you need to keep it covered once it is boiled.
You could also look into the slow chill guys (can't remember their name) which basically they put it into the bucket hot and let it sit until the next day and pitch yeast. IF your sanitation is good, and the bucket covered, this is ok. If not, well see prior problems.
My wife for our copper anniversary got me a wort chiller (#7) this year is aluminum... And it is nice, if he really loves you, he will set you up with a hook up in the kitchen. That part I had to do on my own.
Also if you are doing partial boils, you can pre boil and then cool your top off water and pour that in from the fridge, although you have to make sure your top off water container is sanitized inside. If you fill it with boiling water that will probably be good enough.