It is declared that Howard works for NASA and is on loan to CalTech. However, Howard was employed by CalTech long before he did anything with NASA. He had his job there both before and after he went to space, as well as during all the preliminary preparation (such as his training).
Sheldon claims that he made his first contract in kindergarten. However in the TV show Young Sheldon he makes his first contract when he was about 11 and he made one when his Memaw was dating his college professor.