Ditto on the manual reading. First thing I did when I got the car home was pull the manual out of the car and sit in the house and read for about...oh, three hours. I practically cover-to-covered the manual, and you learn quite a bit from doing so. Whenever you don't know something, something seems weird, or you need to know some info about the car (like what replacement light bulbs to use, or how many quarts of oil it takes...you may be surprised), the owner's manual should be your first reference and line of defense.
Just to clarify, while I don't believe the key for your '03 should be all that different from the one for my '97, if you've only got two buttons on there, one's for the door locks & alarm, and the other is to open the rear trunk. The front trunk can only be opened by having the driver's side door open and lifting the front lever located next to the seat. The symbols for what they do should be on the buttons (a key and a picture of a car with the trunk open), unless they've been worn out and you can't make it out. In that case, the bigger button's the lock button. If you've got more than two buttons on there, then I dunno bout that, but in any case, you may want to try replacing the battery in the key and try then to see if the buttons work then. If they don't, you may want to get it checked out by a dealer to make sure the key is working properly, but I'm figuring the battery's juiced out and the seller thought it was too much trouble to pop in a new one for the new owner.
From the dealer, and again all this is in the manual, the alarm is designed to arm about a minute after the key is removed from the ignition and you exit the vehicle, provided you don't open any doors or trunks after that. Or let's say you opened the trunk, which will unlock the doors as well, and then close it. Again, about a minute later, the doors will lock and the alarm will set. Also, if you unlock the car but do not open the door and get in, the doors will lock and the alarm will reset. How long it takes can be adjusted by a dealer to anywhere from 4 seconds to about 2 minutes, I believe. But you can arm the alarm yourself with the key, either by pressing the lock button or using the key in the door. I wouldn't make it a habit of letting the car wait to lock itself. Find out what's up with that remote, and use that or use the key in the door to lock the car and set the alarm. To go with what JTP was saying, if the horn beeps more than once, something's not completely closed, so the car will not lock and the alarm will not set until you close it (this can inlcude your gas filler door, which is locked when the car is locked). If you get one beep and two light flashes, you're good. And yes, the alarm will still set with the windows down, so keep that in mind if you lock it and then go reaching in for something later on.
For the seat height, once again in your manual as you saw, that requires you to physically move your body in the seat to adjust it. You can lift the lever, but until you actually lift yourself and/or press back down on the seat, the height of it won't actually change. It's one of the few adjustments the did not motorize, though I don't know if that may have changed by the '03 model (doubt it). So give that another try and figure out what height you like for the seat. I'm not a tall person at all, and I actually slammed mine all the way down to give myself more leg room between my knees and the wheel, and I can still see out of the front window just fine.
Have no fear about the long posts and being an excited new owner, I'm sure plenty of people on here can relate, myself included. If anything I'll apologize to you for being a little too wordy myself, but I figured I'd try and give you some explanation along with the advice to reference the manual rather than just tell you to look it up and figure it out. Congratulations, and make sure you pick the brains of the people on here, there's a lot of very knowledgeable folks who do 95% or more of the work on these cars themselves.