Since you've tried the over the counter method, I would agree with Bobbi. Hire a pro. Those sinks are so pretty refinished to their original state and a pro will give you a finish that will last the rest of your life. If you can't find anyone listed in the phone book, ask around at a hardware store.

Temporarily you can lay a thin bead of caulking it will come off both the fridge and counter cleanly. Not seeing the issue and possible space it's difficult to advise you further. Possible to take a picture?

Stand, place your arms *straight* in front of you, as if they were in a sink, so ....a bit like you are ready to dive into a pool. Get someone to measure the height of your finger tips from the floor. This will be the height of the base inside your sink. Once you find the sink you want you'll know how deep your sink is and be able to work out how high to place it.

Nickel can patina over time and I don't often like the look of the bronze with a stainless steel sink. I'd go with brushed stainless steel or chrome.