Pareja de Hecho in Madrid

My partner (EU citizen) and I (US citizen) are planning on doing Pareja de Hecho in Madrid when my current study visa expires in June. This past May we empadronamiento together at his apartment so theoretically by June we should be able to apply for Pareja de Hecho right?

I’m also wondering if I am allowed to be in the US over the summer for work while the process is happening or if it’s recommended to stay in Madrid until everything is sorted - although I have no idea how the process works or how long it will take so I don’t want to be in limbo indefinitely.

Any advice or experiences would be great appreciated!