Learn Spanish (Latin America): Ella nos presta el carro.

In Spanish, object pronouns like nos (us / to us) usually go immediately before the conjugated verb. So instead of saying "She lends to us" at the end, the Spanish word order is literally "She to us lends" (Ella nos presta).

Is nos a direct or an indirect object in this sentence?

It is an indirect object. The direct object is the actual thing being lent: el carro (the car). The indirect object is who receives it: nos (to us). Verbs of transfer like prestar (to lend) and dar (to give) almost always use indirect object pronouns for the person receiving the item.

Could I just say Ella presta el carro a nosotros instead?

You can add a nosotros at the end of the sentence for emphasis, but you must still keep the nos before the verb. So you could say Ella nos presta el carro a nosotros, but you cannot drop the nos. Unlike English, Spanish requires the indirect object pronoun even if you specify exactly who it is later!

I learned that "car" is coche or auto. Why use carro here?