It sounds like it has something to do with the ac compressor which cycles on and off like that. Stick your head in the engine pay and see if that is the case. The compressor will likely be at the front of the block, near the bottom.

The inside of a filter is commonly a paper material. Even if water did not get in, moisture does.

Even if you have legal liability, you will not be sued. Given that they were not fitted they can only come after you for the cost of the tires (or 1 tire).

It seems in any court case it would be your word against theirs. The cost of taking you to court will almost certainly dissuade them from taking that approach. They could use small claims court but again the damage is limited to what they paid you for the tires. It will cost them anywhere between $30 and $100 to file and then they have to consider time of work for the small claim process.

Not a lawyer but if they go ahead and use that tire and it ultimately fails full liability is with them because they knew it was damaged.

Finally given that this was a visible nick, I am pretty sure the onus is on them to find it during purchase. Used goods are purchased as is unless a warranty is provided.

All that to say I personally would lose no sleep over it.

Not big enough to matter but try and bend it open.