Chain on singlespeed tight, pedals won't turn backwards very freely - is this an issue? A well-known and well regarded shop fixed something on my converted single-speed bicycle. On the ride back from the shop, the pedaling felt "different", and I noticed that the pedals will stop after less than a turn when pedaling backwards.

I went back and the mechanic explained that he overtightens the rear wheel in the dropouts. In his experience, within a few days the chain will be back to normal and it'll be fine.

However I wonder if this could cause issues of wear and tear and ruin other, perfectly working, parts?
Yes, overly tight chain does accelerate wear on other parts of the drivetrain. The rear hub bearings are not designed to work under the load described here and if ridden like that for extended periods, it will likely show significant wear.

Chain wear would (and will) cause your chain to gain that desired slack again, but I wouldn't rely on it happening soon enough. Your mechanic supposedly does this to avoid situations where their customers start dropping their chains after their chains have elongated significantly, but from your perspective the trade-off is pointless: setting the correct chain tension is fairly trivial job that needs to be done again anyways if you are to remove the rear wheel for any reason, such as fixing a flat tire or packing the bike up for transportation.

I suggest you ask the mechanic to set the chain to appropriate tightness, or learn how to do it yourself.