33/F ISO pet friendly housing near Stanford or Caltrain from 5/1

My 7 year old Boston Terrier and I will be looking for a new place starting May 1st. My current lease will go month to month at that point, so I'm flexible on move in date.

I work at Stanford so would prefer to be close (< 10 miles) to campus OR within walking distance of a Caltrain stop. I'm open to:
*Renting a room in a place, as long as I have a private bath and there's only one or two other roommates (females only)
*A private studio/apartment/cottage (must have FULLY equipped kitchen) that is unfurnished
*Teaming up with someone also looking for a place and splitting a new lease - especially in a place like Stanford West!

I will of course need a place that's pet friendly (no cats living there, maybe another friendly dog), would prefer ground level with direct access to outside/yard for my pup, and an in-unit washer and dryer (or shared OK, if it's a cottage/studio sharing with a main house type situation). I have a car so need dedicated/off-street parking. Willing to flex on my budget depending on the place, but I top out around $2,200/mo.

Thanks! :)