When I change the FPS in fraps to 30, my recording FPS becomes 30, and when I make it 60, my recording FPS is 60 FPS. However, I was wondering; does raising the recording framerate make the video go faster in the end? I mean, will it speed up my video? (as in, 2 times speed)
No, the FPS setting only sets how many frames per second Fraps will capture. The "speed" remains totally unaffected. Note that if you're not getting 60 FPS you may be making your video file larger than necessary (and taxing an already resource-hungry system). I've recorded in both modes, recording FPS never makes the video "faster" and is actually independent of the actual FPS the game is getting (if the game is getting 10 FPS fraps will simply record the same frames 6 times unless you drop frames).
Using 60 FPS to record is generally preferable, but note it will roughly double the file size over 30FPS and many video software suites and websites don't actually support/play back 30 FPS video anyway. I do not believe Youtube generally supports 60 FPS videos for instance.
If Fraps is like any sane recording software, no.
FPS stands for frames-per-second, and when initially recording a video it's a measure of how many still images the software/camera will capture per second. You're not condensing the playing time, you're taking more images for each second.
Depending on the frame rate that your playback device (TV, monitor, etc.) runs at, higher FPS may look better - although as I understand FRAPS is specifically designed to record your screen or a game, it can only record the rate that your games are playing at (so it will record in 60fps but as it may be recording something playing at a lower fps, it won't look any better).
It will, however, dramatically increase the size of your recorded files and slow down your load times.
I haven't used Fraps but in theory no, because the game would be running on its rate or fps. You're only modifying the rate at which Fraps records the game. So let's take for example Street Fighter 4 which runs at 60fps. Recording at 30fps means you'll miss half of the frames (i.e. looks choppy). Recording at 60fps means in theory you're capturing every single frame the game is running (i.e. smooth). Now recording at 120fps for a 60fps game should not get you any additional benefit.