are there any animals alive today that survived the permian extinction and have remained relatively unchanged?

Don't assume that because an animal looked the same then than it does now that means it has not changed. There was a lizard in the Cretaceous which looks identical to ancestors in Australia, but internally the animal changed a lot, on both a physical and biological level, but its outer appearance seems the same. 
 

With that said, many arthropods and cephalopods look the same. Check out the horse shoe crab 

Protozoans, sponges, jellyfish . . . .

Sharks are relatively unchanged over eons. They do very well as is.

No.  Our oldest animal species are from the Triassic.  Change happens.
Edit:
"Nearly identical" is not the same.  "Horseshoe crab" isn't a species.  Neither is "nautilus."
Triops cancriformis -- down to the species -- have been extant from Triassic to today.  That's the oldest animal species.