One of the good things about GT is you can go at your own pace.

Don't worry about missing tutorials. I don't think they're removed after a time, although some seem to get updated with improved versions.

Songs are sometimes removed, but that's only because of licence renewal issues.

I think moving on is good as long as you include practicing the things you're still struggling to play at speed. You'll probably know if you've gone too far because things in the lesson will seem impossible because you can't do a simpler version or similar technique that was in an earlier lesson. Another way of knowing is if you suddenly feel bombarded with new things.

As for increasing speed, try using a metronome at the speed you feel comfortable at, then try increasing it a little, such as anywhere between 2 and 5 BPM faster. Keep doing this until it's too fast, then go back to slightly slower than your fastest comfortable speed. Make a note of that, then try starting at that speed the next time you practice whatever that is. If you find it's too fast, then try a slower speed. It takes time to build up speed.

Another thing to try is make the movements very slowly and look at the amount of movement each finger makes. For example, changing chords - could you make the same change but not lift your fingers so far from the fretboard. You might find some of your changes are being slowed down simply because your fingers are moving farther than necessary away from the fretboard.