Hi ! I just saw that we can't use an emulator, and, how could I do to make a speedrun of the demo version? which is complicated to find in cartridge. Otherwise, I have an R4, but I do not know if it's considered an emulator or not ...

R4 is ok, Find the kiosk version on internet and put it on your r4

OK thank you but I have a quick question: what can I do to directly film the screen of my DS without going through my phone and have a bad video quality?

Buy a capture card

There is a capture card for the original ds but it is quite expensive. You can find it here: http://3dscapture.com/ds/

It's quite expensive, so it's OK to film with his phone ? :/

Yes

As long as the video looks fine I think it is ok.

OK thanks :D

Yes, as others have said, flashcards are fine, and yes, you can use a phone or webcam or whatever to record as long as the game is viewable