Mentors advocating for you?

Hi all, have two questions for you, would greatly appreciate any advice!

1. How acceptable/common is it to ask a mentor (at an outside institution) to advocate for you at that institution, if say you only worked with them on one project, and maybe met over Zoom like once, but their vibe seemed to be pleasant?

2. I've heard advocating helps most only if you didn't get an interview (e.g. the advocating gets you an interview). So does that mean we should reach out to the mentor only after the last day passes where their institution is known to send interviews (e.g. if they release interviews intermittently over a 3 week period, send your mentor an email only after the 3rd week)?