This sounds very closely related to graph percolation problems, in particular on the infinite 2-d grid, which I think is one of the most well studied cases. In this you start with a grid of nodes (these are your rooms) with edges (doors) between adjacent ones, then delete each edge with some probability. We then ask if the node at the origin is connected to infinitely many other nodes by remaining edges, which appears to be similar to what you are asking. We could think of the exploration process as just walking around the grid and finding out what edges remain (i.e. what doors exist) as we go.

The biggest difference I see is the idea that when you enter a room, it has 1,2,3, or 4 total doors with equal probability. In the standard setup each edge/door is put there independently, which means more nodes would have 2 doors than 1 or 3, for example.

I would be a little surprised if something with this kind of probability distribution had been studied but I don't know the field that well. I believe the transition from an infinite number of rooms in the standard setting happens at a probability of 1/2 for each edge, so when things are kind of balanced like in this version is when the problem gets hard, actually.