Je Tsongkhapa had teachers from many different lineages. He made a lot of effort to compile the teachings clearly and comprehensively. Gelukpas have put a lot of effort into keeping things in order.
The difference between schools is over exaggerated. If you look at the lineage lines of transmissions they go through a lot of the same people. And then practically, you look at the lineage holders and they've received teachings from people from different schools. If there was some big conflict in the instructions they wouldn't be able to do that.
Yes, but the book is describing Indian Buddhist tenets (Vaibhasika, Sautrantika, Chittamatra and Madhyamika), not Tibetan schools (Nyingma, Kagyu, Sakya, Gelug).