Can I fix these iPhone glitches?

For the past three or four days, I've been having the following problems with my iPhone 4s:
1. When I take my phone off a speaker, it seems to be stuck in the speaker audio setting. It doesn't play sound or give me any volume options. I have to put it on and take it off the speaker a few times for it to return to normal.
2. My music stops mid-song but the iPod app is still open (it doesn't crash or anything).
3. My apps have been crashing more frequently than usual.

The only reason I can think that could be causing this is that I dropped my phone about a week ago without a case and the back shattered. I wouldn't think that this would cause internal problems though. And yes, to answer your question, I have restarted the phone many times.

Is there any obvious or common issue that could be causing this?

Your question is like a patient asking what caused his cancer, could it have been the microwave he's been using all his life or the cigarettes he's been smoking for 30 years or genetics?

The answer is both or neither or either one, you just can't tell and now will never know.

Yes it is POSSIBLE that your iPhone was already defective and the drop did nothing, or the Dropping of it loosen pieces on the inside, (Oh yes it happens to all electronics, happened to my laptop too when I shut my lid to hard, knocked the processor right off it's place)

The answer is irrelevant now though, Once you've dropped your phone and caused physical damage, your warranty is gone. You could call your service provider's customer care and scream until you're blue in the face that you were having problems before the phone ever hit the ground, but it doesn't matter, it's broken now, it's your word against theirs and if they see it's broken, they're not going to try and fix it.

Your next thing is you need to see if you have insurance on your phone line to replace the phone. It's an additional fee on your bill so if you're not sure, take a look at your last bill, you don't HAVE to have it, and most people choose not to get it, but when this situation occurs, you'll wish you did have it, but now you won't be able to get it.

So then check your upgrade, see when it is, if it's a long ways off, again, don't bother calling customer care, they CAN'T change it (I worked for AT&T for 2 years, I know how it works and iPhones are off limits when it comes early upgrades) they don't even show up on their screens to be sold!

So the last step is to look elsewhere, like eBay, and then sell your old one on eBay (People will still pay money for a broken iPhone because they want the parts)