Hi, JyLiss!
Ok, cool. We've established that we have the same system. This is good, because I have the *exact* same problems you're talking about. I have an idea to figure out how to make it stop. First, let's go through your other topics:
[quote author=JyLiss link=topic=69675.msg1247899#msg1247899 date=1173033271]
They also made me pay for an extra key because they said that a key gets integrated into the remote start system. [/quote]
I want to make it clear that they have lied to you. I've gone through two installs of this system. One by a dealer, and one I installed myself. I've read through the owner's guide, installation manual, and done every painstaking step myself. This is a complete lie. It makes me mad because it does, indeed, sound true. It's natural to think "How else can a car start? You must have to put a key in there somewhere." This is not your fault, they told you a very cleverly-crafted lie. But this makes me very, very angry.
[quote author=JyLiss link=topic=69675.msg1247899#msg1247899 date=1173033271]
There's supposed to be a reset switch on the door, but there's only this black rubber foot type thing on the bottom left of the inside driver's door. It does not depress when pushed. Have you found out anything? - JyLiss
[/quote]
DreamWave is correct on this. This is an override button. It should be where your left foot is, but NOT on the door. It is a button surrounded by a frame so that you don't accidentally press it.
Now: let's talk about the remote start problems.
1. "I set off my car alarm periodically for no reason. I unlock the car with the remote, get in, and all hell breaks loose."
This is the more annoying problem, because it wakes everyone else up in the neighborhood.

I've tried two techniques to avoid this, and one of them has (so far) stopped the alarm from going off. Let me guess: it usually happens when you get into the car, right? And, it doesn't matter if the car is running (with remote start) or not running, right?
Here's my technique #1: Unlock all doors, every time, and slowly. To unlock all doors : always unlock, then wait one second, and then unlock again. I always wait a second before clicking again. I also don't open the door within 2 seconds of unlocking it. There are two points: Don't rush things too much, and unlock all the doors (press unlock twice).
See if this works; if it doesn't, then we'll try my other technique.
Problem 2: "other times I hit the lock button and then hit the starter button only to get three blinks of my lights accompanied by short little horn beeps"
This is almost certainly due to a door ajar problem. You've probably left a door, your trunk, or (less likely) your hood open slightly. Commonly, for me, it's the trunk, because it's hard to tell when it isn't quite shut.
Did the dealership give you the owner's manual for the remote start system? If not, I can probably scan mine and send it to you.