A dog does not need to be purebred, or a GSD, to do SchH at the club level. Registered purebred status is only required at some of the higher levels. Any dog with the suitable temperament is welcome to participate.
As far as the "temperament test" there are 2 different ones in a trial and I'm not sure which you are referring to. What is usually called the "temperament test" occurs at the beginning of the trial where the judge will have everyone participating get their dogs out and walk them around, and will also check ID (tattoo or microchip). They are looking for a dog who shows confidence and no aggression toward people or the other dogs. That's it.
The other "temperament test" would be the BH, which has 2 sections. There is an obedience portion that consists of on and off lead heeling, sit and down in motion, recall and finish and then a long down stay. Then there is the traffic test where the dog is exposed to people, dogs, bikers, joggers, cars and must remain sound and neutral. Dogs must pass the BH before going on to SchH/IPO titles. The details of what the BH entails, as well as the other titles, can be found by looking up the IPO rules (which are available online).
Now if by "temperament test" you mean an evaluation by a club/trainer to see if the dog is suitable for participating in SchH rather than the temperament tests in an actual trial, that will vary from club to club but usually involves seeing how the dog is around people and other dogs (similar to the temperament test before a trial) and then also testing the dogs food, prey, and possibly defense drive to see if the dog has the temperament traits necessary to participate in SchH.