If one says "han er ude at købe" without the ind, people will be waiting to hear what he is buying.
"At købe ind" is just another term for buying food/groceries at the store, and I have only ever heard it used for that, so including food in the answer sort of clarifies it. I don't know where the "ind" comes from, though.
"Hun er ude at løbe" = "she is out running (lit.: to run)" etc, so I think you're right. I assume it would translate to the progressive verb at least most of the time, since "er ude at" always means that the subject is doing something right now while being away from home or the house.