Uriel did not get cold the way humans did, since his body temperature stayed constant most of the time (an added benefit to having flames in his wings). But he could feel the cold, and he did not like it. He probably would not have gone and found a new heater himself, but Eve had offered to take him, and he would never turn down a chance to spend time with her.
It was just shopping. It was not inappropriate for him to be doing.
He opened the door when she knocked and finished wrapping a scarf around his neck tightly. His apartment looked like a book hoarder had very meticulously organized their collection. There was not much in the way of furniture, and every flat surface was covered in books of some kind, or speakers.