Are there aged cheeses I can make which don't require a cheese fridge?
The cheeses that just plain won't work in the fridge are those that make use of bacteria or mold with "special needs". Swiss, camembert, and any blue cheese come to mind. The bacteria that produce the holes in Swiss need specific, relatively warm, temps for specific times. Likewise the molds on the rinds or through blue cheeses need warmer but still cooler than room temperature.

Most semi-firm to firm cheeses can be aged just fine in a fridge. Many generic cheese recipes such as:
http://biology.clc.uc.edu/fankhauser/Cheese/Cheese98.htm
or
http://www.cheesemakingrecipe.com/Dutch.html
either say to age in the fridge or don't give a temperature at all and just say "age in a cool place".

In particular cheddar or gouda like cheeses age just fine in the fridge. Parmesan should also work but I personally haven't had the patience to wait the year necessary to try it.

Of course if generic-cheddar-like cheese isn't good enough and you want to recreate some particular cheddar that will be harder. Those recipes usually call for something like "Age for 6 months in this particular cave in France".

As for room temperature. I don't know. Room temperature at my house is 80F and no cheese will work with that.