How long will this cake survive on the counter or in the fridge?

Depending on the ambient temperature of the place you're keeping this cake I would not suggest keeping it at room temp for more than 24 hours after it's come out of the oven. Before the apples ferment they will go "off" by getting slimy and could make someone ill. But in the fridge it should keep at least 5 days, maybe 7 if kept tightly wrapped. Might be nice to toast a piece before eating if it's gotten too chilly in fridge.

I have an apple cake recipe that is similar to this one in the sense that it contains an oil-based batter and fresh apples layered in between the batter. That cake, and I would say this one too, can safely sit on the counter for a day or two. After that, we would refrigerate it over night and pull it out as needed or a bit before to come to room temp. The worry is that the apples will eventually begin to ferment.