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: 
For lighter beers I usually try to store as many in the fridge as possible because the longer fridge time will help them clarify in the bottle before you open them (if that is pertinent to the style and/or important to you). Of course this is after at least 2 weeks of priming time outside the fridge at about 70 degs.

I have also had some brews that maybe I bottled too early or otherwise I noticed they were overcarbonated or had a bottle explode. For these types I would definitely rush them all into the fridge to stop the yeast from carbonating any further.
I ferment ales until they stop making gas, typically 8-16 days. After that I leave them for around 3 weeks, 70 degrees, in the bottle, in the dark, upright. Never a problem!