You can't officially rent an apartment without a residency permit of some sort. Renting a room is a way around that. Sometimes someone blocks off one room in an apartment for their own use, then rents out the rest of the apartment as a 'room', in order to get around all the restrictions.

I heard recently that the few really cheap backpacker dorms are now filling up with Indonesian workers, and that the backpackers have had to migrate to the more expensive ones.

I think Singapore still requires that you leave the country to get your work permit, but I don't think they require you to return to your home country.