Is it okay that I turned on my new fridge immediately after delivery?
It is normal for fridges to make all sorts of sounds. Some of it will be materials contracting, the compressor running, etc.  If you have a water dispenser or ice maker, you will occasionally hear water flowing.

You are supposed to let a fridge sit a bit after delivery, but most people will just plug them in.

I doubt you have damaged your fridge.  Plug it back in, let it cool down for 24hours, and assuming it keeps working, you're in the clear.