You are correct on your assumption of the difference between class and type. I teach and examine on type rated aircraft at work and SE/ME class rated aircraft in my spare time.
It is my understanding that you need to have 10hrs on each difference you wish to teach as well so 10hrs for tailwheel, turbo etc.
Yes, the differences sign off is carried from your own sign off. You don't need to go and get it again as an Instructor.
I certainly think trying to teach a difference that you have no experience off will leave you in hot water if something goes wrong.