I record the speedrun on any screen recorder (like bandicam) and just put it on youtube? Or do me need to do something else? and how to apply for speedrun consideration?

yeah record it with what you can (id recommend OBS over bandicam but it doesnt matter) and upload the video to youtube. to submit run go to a games leaderboard and hit the "submit" button and fill in the form

Could I cut or increase speed at uninteresting moments?

If you plan on submitting the run to a leaderboard, no.