Should I notation my school chess tournament?

Not trying to be arrogant, but I BELIEVE I can write down the moves of the game while still being able to go 7-0 :)
Do you guys recommend taking notation of all 7 rounds?

It's not a very arrogant statement (depending on the opposition) - but sure, you should record all of your games as soon as you are comfortable doing so. 20+0 is loads of time.

if you can, you should go for it. Maybe the opponents feel that is weird if it is not common, but for you it will be a nice tool for analisys. You can save the games in for example and have all your games collection and analisys in one convenient place.

It seems strange that you would recommend another website that does exactly the same thing as Lichess. Do you work for chess microbase?