I have an electronic drumset and some guitars/AXE-FX2 which I usually play through headphones...but would it sound much better if I got some studio monitors? It would be for a small apartment bedroom, about 14x14 feet, and I probably wouldn't blast it at full volume so I don't bother the neighbors too much. Can you recommend me a good pair of studio monitors, and tell me if you think it would be better than playing through headphones? I don't mix music much...just play live. Thanks. As a side question...are studio monitors good to use for PC gaming as well?
If you're not mixing, then I'd say just use headphones. If you're only playing in a small 14x14 area and can't turn it up, speakers would be a waste. Also, you have to remember that an amp flavors the tone of a guitar signal. I much prefer playing through an amp, as opposed to playing through speakers.

Most studio monitors are a little light on the bottom end. Will they work for computer monitors? Sure. Depending on what you buy, you may or may not like the results.
Your bass drums would sound better on a bigger speaker. I personally prefer a 10" powered speaker. I use one for my yamaha dd-55.
First, about the headphones. If you're using cheap four buck buds, your Axe isn't going to sound very good. I recommend spending $100 to $200 on a set of real headphones -- something like the Sony 7506, AKG 240, Sennheiser 280, Beyerdynamic 770s.

Your monitor speakers will depend on your budget. I'm using a pair of powered (about 80-90W) KRK Rokit 8's. They're designed (as are most monitors) for nearfield use, meaning that 6-8' from your ears is their sweet spot. These have 8" woofers and plenty of bass response. Understand that it's not speaker size that determines how low the speaker will go -- in the bass region, the size of the speaker mostly decides how loudly the speaker will reproduce that bass. I chose the Rokit 8's because they produce the lows loudly enough that I don't have to spend the extra bucks for a subwoofer, where IMHO, if I'd chosen the Rokit 5's (for example), I would have eventually wanted that sub.

And yes, you can use powered monitors for recording monitors, gaming, TV and for reproducing your iPod.
Some monitors I like that won't break the bank:
KRK Rokit 6
KRK Rokit 8
M-Audio AV40
They are excellent for guitar and decent for general mixing/listening. They are very boundary dependent so you need to move them around a bit from a back wall to get the best sound.
"Your sound is in your hands as much as anything. It's the way you pick, and the way you hold the guitar, more than it is the amp or the guitar you use." -- Stevie Ray Vaughan

"Anybody can play. The note is only 20 percent. The attitude of the motherfucker who plays it is 80 percent." -- Miles Davis

Guthrie on tone: https://www.youtube.com/watch?v=zmohdG9lLqY