hello! Where should the development log be kept?
Should the devlog be started already or only from May 20th?

You are allowed to work on your game and/or Devlog how much you want, however, you must submit it by the deadline. No exceptions.

Itch.io has a Devlog feature located on the game edit page. You can start one and save it there, add to it everything that you worked on then when you are ready to submit just publish it and then fill out the required fields in the Game Jam game submission form.