I'm just wondering how can I speedrun new game that's not on the site

Being technical and literal, you can speedrun any game, it doesn’t need to be on the site.

If you want to request a game for the site, you can do so here (please make sure to read all rules before submitting) http://www.speedrun.com/requestgame

If you can finish it in one sitting, you can speedrun it.

Some extremely long games even have break periods written into their rules, though this should only be for extremely long games and should be avoided at all reasonable costs. Any run under 10 hours can be done in one session, a 20 or 30 hour run....well it can but it probably shouldn't. The in-between is a bit of a grey area.

I had to speedrun a game but it needs to be 7 days old. I need to make a rule of my speed run first.

I mean it doesn't really need to be done in one sitting, there's ILs runs for some games. Some even permit splitting in the run cause the game is too long (even like 700 hours) to be done in one sitting :3

Where are the rules for submitting a new game?

Rules for submitting a new game are here https://www.speedrun.com/requestgame?series=