So basically, when I RAM Preview, the timeline skips a bunch of frames, and the video plays fine. I've tried everything I can think of.
Try hitting zero on the numeric keypad and see what happens. You're just hitting the most worthless button in AE, the Play button. Besides, how can you TELL frames are skipping? There's no video in that comp, just an audio track! Furthermore, you you don't show the Info pane, which displays the actual frame rate of the RAM Preview, so you have no authoritative way to determine if AE's skipping frames.
Oh, well. If the problem persists, you should give us more information.
It probably wouldn't hurt you to look at this as well. After effects ain't like a smart phone app, y'know. It takes some knowledge of the basics, and as far as we know, you're just a raw rookie.
I watched your video about 3 times. I have no idea how to tell if the video is skipping frames because I don't know what the original video looks like. If you step through the ram preview one frame at a time using the page down key are you missing frames? If so, go to Edit>Purge>All Memory
Then go to Preferences/Media & Disk Cache and Empty both caches.
If that doesn't fix the problem I think the codec you're using for your screen capture would be the culprit. Try recording your screen cap to a different format.