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: 
You'll only get bottle bombs if you over-prime or bottle too early (or have an infection). The difference between fridge, celler and even hot (80-100+°F) temps will not cause a normally-carbonated beer to explode, or it'd be happening commercially all the time.

That being said, heat and temp fluctuations aren't ideal for your beer.

Make sure fermentation is finished when you bottle. Don't over-prime. Be clean. Then just keep the bottles in the coldest darkest place you have access to.
if you put the beer in the fridge after a week of bottling it will be a little flat but still it will taste better than cheap bear from the store . if you want to be safe wait 2 weeks befor putting in the fridge