So, I know why I have this error.
My rule on firestore:
allow read, write: if request.auth.uid != null;}.
So, if my user is connected, i can use firestore correctly.
But how to do if it is a new user without firestore/auth account to use firestore (I need to use firestore firstly to be sure the email is not taken.)?
Firebase gives you the ability to log users in anonymously. The documentation for this is found here. I would recommend logging users in anonymously, allowing them to provide whatever data you need to create their account (email in this case), performing whatever logic you need to do (checking if the email is available), and then turning their anonymous account into a permeant one (this use-case is also documented in the link I gave previously).