Cheese – use Cheddar cheese for pizza

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.

This is a combination of the type of cheese and too much heat. Some cheeses melt more readily (mozzarella for example), but all of them will seize up if they are heated too much too fast - the proteins 'curl up' and separate from the fat and water in the cheese.

To combat this, you should chop or grate the cheese up to speed up melting, and lower the heat a little before adding it. You should also avoid using cheese straight from the fridge.