Did the T-Rex evolve from a plant ....? Did animals ever live on earth without plant life first...?

Animal life is not possible without organisms that do photosynthesis, there would be no free oxygen without photosynthesis. Before photosynthesis the only life there was were peculiar forms of bacteria which used sulphur as the oxidizing agent. An animal that is capable of photosynthesis would not be "part plant-part animal", it would be an animal that is capable of photosynthesis. Once you understand mitochondria the concept of chloroplasts in animal cells is not that strange.