I'm sure this is addressed elsewhere in the forums, and to a much better extent than I provide here, but here's some quick info that might help:
People add hot wort to ice all the time to cool down the wort. It shouldn't hurt your beer.
It sounds like you plan to put the boil pot into an ice bath in your sink and add ice to the boil pot. I doubt this is most efficient, because the boil pot holds a lot of residual heat which will tear through that ice quickly. Instead, you should aim to get your wort out of the boil pot asap. The problem is that people say you shouldn't add hot wort directly to a plastic fermenter, even if it is food grade. That's why you don't just pour boiling wort into your fermenter. If you're fermenting in glass, i'd imagine too hot of wort would crack the glass.
So, I think the more common practice is to put the boil pot into an ice bath in the sink like you plan, then maybe circulate the ice water around the boil pot in the sink for a few minutes, which will quickly reduce the temp from 212+ F. I don't have a lot of experience doing this because I do have a chiller, but I'd say at least 10 mins in the ice bath circulating the water around the pot should cool it down significantly. Meanwhile, add the ice to the (room temp) fermentation vessel like your plastic bucket, then add the wort to the fermenter. That way, the ice isn't battling to overcome the residual heat from the boil pot if you were to add the ice right to the pot. I believe that calculating final temp when you add hot to cold is a straightforward linear weighted calculation (equal amounts of 33F water and 133F wort should put you at 83F temp combined, I believe).