A lot of newer tires are getting wear bars that are thicker than 2/32. In PA the official inspection law is two consecutive treads of 2/32 or the wear bars, which ever is lower.
As for minimum to get them mounted, that all depends on the shop. I once told one of our broke customers that he desperately needed tires. He brought us some new to him junk yard tires to put on. I objected to the boss saying that the ones he brought us were worse than the ones he already had, but was told to put them on anyway.
I did note my objections in writing. If you tell the shop that you just need them for rollers to push it around the yard or something you could probably get them to mount Mexican racing slicks as long as you pay.