First I have to say that Both Quests caught my attention and I decided to try, took a while and I played it safe but now I wonder if I can cut the video to only the Second Quest part and submit it as Second Quest. I'm no saying submitting the same video for both, There will be one full video of Both Quests and other video with only the Second Quest.

Can I do that?

If not, how do I start the game in second quest? I've seen some runs that immediately start on second quest and as far as I know you can only do that by beating the first.

No, since in Both Quests you start from 8-1 of the second quest, while in Second Quest, you start from 1-1.

I started Both Quests from 1-1... I didn't read that Both Quests allows using B to start from 8-1 so I ran Both Quests starting both from 1-1.

in that case yeah you should be fine