I got an offer for a tenure-track assistant professor position in USA, Department of Engineering. The university national ranking is between 35th and 65th at the US News website. if I will have any left start-up money for equipment, could it be used to pay PhD students or my summer salary? Or are they two different accounts? Does it maybe depend from University and University?
This varies not only from university to university but from department to department. In some places it wouldn't be a problem, and in others it may even be impossible.
I recommend you get in touch with whoever your point of contact is (whoever extended you the offer) and ask them directly. It is important that they know you want to do this as this will naturally affect how much startup funds you are liable to need.