To me, the spots in the pan look like harmless water spots but the dark gray residue on the paper towels looks like more of an issue.
Water contains numerous minerals and particulates. When you heat the water, these contaminants can leave spots on the surface of the heating vessel. This can commonly be seen inside kettles used to heat water, where there is often a residue of calcium carbonate build-up. It's nothing to worry about. It can make your water taste bad but other than that, it's only really a cosmetic issue.
The gray residue is a different matter. If you're sure your pot is stainless steel, as opposed to aluminium (in which case it could be aluminium oxide), then I'd suspect that it's a residue left over from the final polishing step of the manufacturing process. My advice would be to wash the pot on warm soapy water and then try wiping it again with some paper towel. If you get the same residue, repeat the washing process.
If repeated washes don't make a difference, then that would seam to indicate that it's some kind of oxidation or mineral deposit that's forming. Something like Bar Keeper's Friend should help in that situation but I've also read that certain cleaning products and sponges can actually cause oxidation. So you could also try using a different type of sponge or cloth to wash the pot. If the residue disappears at first but comes back at a later date, that would also indicate that it's oxidation.
Having just done a quick search, I've also found this thread that looks like it's relevant to your issue - Stainless Steel Pan -- gray bottom. Why?