So as a cheap winter alternative I picked up some AS Sumitomo HTR+'s for my car from TireRack - got 'em mounted, and 3 days later got a flat.

I ran on the sidewall for a little bit and you can see the text that's usually on the sidewall is worn down. So far it doesn't look like there's any damage from foreign objects like nails etc, it might have just been a bad mount?

My question is, with the sidewall damage, could I file a complaint against the guys who mounted my tires and ask that they replace my tires?

 

The tire might look ok externally, however, if run on flat for any period of time the internal structure can be damaged. The plys in the sidewall and tread area could be damaged. Also, the innerliner of the tire could be torn which will make the tire leak air and lose integrity. Even driving the tire flat for a couple of blocks can cause these problems.

If possible, take the tire back to an installer and have it inspected to see if any of the above has happened. If it didn't happen right after you left the installer, I doubt they will do anything to compensate you. However, it dosen't hurt to ask for compensation, wost they can say is no.