Couple of questions about Naga running.
1) How to make Naga go down quickly after jumping.
2) Are there any specific routes to take in the running sections or does everything take the same time?

1) Use the Naga swipe attack.

2) No specific routes. I suggest choosing one route and sticking with it. There's a trick to save a few seconds overall, though. Turn corners as early as possible. You can turn around corners much earlier than it seems. This gets Korra moving down the next alley that much sooner.

Good luck!

I have a question about Naga. In Tylos' run in NG+ Extreme I see him jumping mid-air without using earth bending, most notably at 5:20.

What exactly is he doing to do that?

When you air boost then jump, you then press the air button again to do that little boost