Question: My instructions that came with my kit said that I should let the bottles sit for 3 days, then age in the fridge for 10 days. If I don't refrigerate, will I get bottle bombs?

Advice on forums and other places says to leave them in the bottle for two weeks or even upwards of a month, I doubt every brewer has enough fridge space to store 5+ gallons of beer for 1 or 2 months. So, I am inclined to think they won't blow, unless I bottle with too much unfermented sugar in them. 

Will leaving them out in room temp, cause them to pop?
Are there any other reasons for me to refrigerate my beer, aside from taste?

Answer: 
My process is to bottle condition at room temperature then every 7 days put a bottle in the fridge overnight and give it the taste test. 

If its carbonated to your liking then load up the fridge, else wait till the following week.
I do the same. Once bottled, I chill/drink no sooner than 10 days, but leave out as many as 4 weeks so far. We've been having a heatwave recently and I'm trying to determine if the bottles are too hot. Probably closer to 80 degrees, but I have no way to cool. I did put the fermenter in a bus tub of water and wrapped with a wet beach towel, but as far as the bottles are concerned, they are on their own.