Cleaning brown/gray spots on overheated stainless steel stock pot

Be careful with cold water in a hot pan - since the pan is metal, you shouldn't have to worry about cracking, but you will get a nice burst of steam in your face!

Your best bet is to deglaze the pan immediately while it's still hot, or heat it back up and deglaze it. There are some tips in this thread. As a bonus you're going to get a delicious pan sauce as part of the deal.

You're never going to get it store-bought perfect, but I've had a lot of luck cleaning stains that seemed permanent by using Bar Keepers Friend, which in my opinion is one of the best commercial cleaners available.

This happens to mine, and they are most definitely stainless, not aluminum. Our stainless sink gets the same way. I assume it's from all the minerals in our water. I have found that mixing up some Oxyclean and water and a couple of minutes' soak and a scrub with a plastic scrubber will get rid of it (both from the pan and from the sink). It comes right back, though.

I haven't found that it has any negative impact on the usability of the pans, and doesn't shorten the life significantly if the 14+ year old chef's pan I use almost daily is any indication.

I did a little more research, and it's definitely mineral haze. Recommended treatments are Barkeeper's Friend (mild oxalic acid) soak/scrub, or boiling some vinegar, either one followed by a thorough rinse. In theory, if the haze is left alone it could develop into buildup, which would begin to degrade performance, but in practice it's not an urgent worry.

So if it bugs you, clean it off with mild acid or oxygen bleach. But expect it to come right back. If you only removed it once a year, you'd be a couple of years ahead of my cleaning schedule, and I haven't seen any harm from it.