Ice boxes were pretty common in many areas, including urban cities like Manhattan until late as the 1950's. Popular way to show/tell early morning was saying or showing the "ice man" with his cart making his morning rounds. Next to the milkman, he was one of the few other "service" men up and at work by the crack of dawn.
One put huge slabs of ice from the ice man on the bottom of the unit, and as it melted, there was a drain pan to catch the water somewhere underneath to catch the water.
Many old apartment buildings in NYC, including some very ritzy ones had "ice boxes" built into the kitchen wall that faced outside. That is there was a small "box" like square with a door that one opened and put things like milk or butter in, then closed the door. Of course this only worked during deep winter months, from say November to March, but long as out door temps stayed below 35F, things would keep.
As for "ghost town" objects, surprised this person was allowed to wonder around such areas and take things as he saw fit. The bit abouts rat's and mouse nests would put me off the thing anyway. Last thing one would want is to get up in the middle of the night for a snack, go into the kitchen and see a "housewife" in turn of the century clothing standing near the fridge, asking what one did with her joint for Sunday's dinner! *LOL*
*Feets Don't Fail Me Now*!
Launderess