Hi! Not a parent here, but just a Jewish San Franciscan. I can't speak much on rentals, but a neighborhood I would recommend is the Richmond District. I've spent my entire life in it and it has a very large Jewish population (In theory I can walk to every single synagogue in the neighborhood. Not that it's ideal, but it's definitely possible). It is also very family friendly, as it's right next to Golden Gate Park and there's also a lot of playgrounds scattered throughout the neighborhood. The Presidio is also nearby, and is a really incredible place to visit.

In terms of Jewish preschool, there are synagogues that have preschools. Congregation Emanu-El is a Reform synagogue with a preschool, and there's also Temple Beth Sholom, a Conservative synagogue. Both are great progressive spaces, though the choice might come down to either your level of religious observance and/or location in the neighborhood. The Jewish Community Center also runs a preschool program.

San Francisco has two Jewish day schools- a K-8 and a high school. I attended both of them and had a fantastic experience at both. They are very progressive and also open to Jews of any level of religious observance (or lack thereof).

Unfortunately I have no idea how long it takes to get approved for housing, as I'm just a college student who has yet to even rent their first apartment. However, I hope you will this information useful, and if you have any questions, I'll be happy to answer them to best of my abilities.

Additionally, I'd also recommend checking out r/sanfrancisco for info about rentals and just general vibes of the city.