Concerning the tabs and the second video example, note that Paul Gilbert spent his whole life perfecting his picking technique, so it shouldn't come as a surprise that playing his stuff is very challenging, especially for the picking hand. For that riff to sound good you definitely need to use alternate picking (as he does), not raking/sweeping. The latter technique can be useful for other things, such as playing arpeggios or fast linear scale runs.
I believe that there's no other way to learn it than the hard way: practice slowly, cleanly, and with a metronome. If possible, play amplified and with the appropriate distortion sound as often as you can, because otherwise you won't learn to achieve the right sound (e.g. when palm muting) and you won't be able to hear any noises caused by other strings etc. It will take a lot of dedication and time. If you have the feeling that you do not progress, have an experienced teacher look at your technique. This can save you a lot of practice time!
One important thing is to experiment with the angle of your pick. Paul Gilbert holds his pick at an angle of almost 45 degrees to the strings. In this way you will feel less resistance when crossing a string. Another thing to try is pick-slanting, i.e. turning the pick around an axis which is parallel to the strings. Troy Grady has a great youtube channel, mainly dedicated to picking technique. I'm sure his videos will help you a lot.