As far as I know, GarageBand doesn't add anything to the end of exported tracks, but it exports as much as there's sound in any track. Therefore, to avoid having extra "stuff" exported, you should always create a loop region and export with that. As you can see in the following picture, I'm at second 47 in the "timeline", however, I've created a region (the yellow bar on top) that starts at second 1 instead of 0, therefore creating a 46 second clip.
The exported version can be seen in quicktime with a length of 46 seconds, without any extra silence or sound.
In order to create such region, you have to activate the "repeat/loop" (next to the Play button in GarageBand). That will create an extra space below the Beat Bar (where you move the Play Head). When you hover your mouse over that small region, the cursor changes. You can click and "drag" to paint the yellow zone. That's the zone that will be repeated over and over.
Always export with a repeat/loop zone created and Garageband will only export that.
NOTE: Logic Pro/Express behaves the same way when creating bounces.
There's a draggable cursor at the end of the timeline on the upper right corner. If it is beyond the end of your sound on your longest track GB will include that much silence. Drag it back to the end of your longest track, or leave however many seconds of silence you want.
Update 2020 - What about iOS?
I don't use macOS anymore but if you want to do this on iOS or iPadOS, if I don't incorrectly recall, your only choice is to have a region that encompasses the part you want to export, tap it (to select it), then tap it again (to see a floating set of buttons), and pick "Loop". I don't know/think there's a way to do the same "playback region selector" on iOS. I may be wrong, but this question was about the OS X version of GarageBand so I don't know more :)
Yup, if you no longer want it on your system, those are the two biggest offenders in terms of size. To be completely thorough, have a look through ~/Library/Preferences/ for the .plist file and trash that too.
If you're on a purge, it may be worth looking at iMovie/iDVD too if you don't use those.
Best Answer
If iLife came pre-installed on your computer back in 2008, your computer came with one or two install discs. If there were two, iLife installers exist on the second disc.
You didn't specify what kind of computer your old or new computers are, but if they both have FireWire ports, you could use Target Disk Mode and a FireWire cable to transfer files faster than via your slow network.
A third method to transfer GarageBand (and other applications) from your old computer to your new would be to use the Migration Assistant application. You can't select individual applications to copy, but you can fairly seamlessly transfer applications from one computer to another, then delete what you don't want.
Without knowing any specifics about hardware or OS version, it's difficult to give you better advice, but your three best bets to more seamlessly copy the application are 1) the install disc, 2) copy the files using Target Disk Mode, or 3) use Migration Assistant.