If I send a book and the person gets it, how long will it stay on my books mailed list if they don't mark it received???

It will remain in your "books mailed" until the person either marks it received or the system marks it "lost"

If your tracking states that it's been delivered, you are permitted to send a polite PM that's along these lines: Hello, I noticed that the tracking on the package for "book title" says that it has been delivered.  Could you mark it received if you have received it?  Thank you."

26 days if it's traveling within the contiguous United States, and 35 days if it's crossing salt water.

If tracking says the book was received, you may pm.  I'd recommended waiting a minimum of 2 or 3 days...I wait a full 7 (IIRC PBS recommends 5 or 7ish).  Not all PO's use the 'delivered' scan correctly, and some are still waiting delivery at the local PO when marked delivered.

Also, sometimes there's a bit of a delay on the receivers end...say they are out of town or having the day from H&ll.  I really try to log my received books in immediately, but once in a great while it's the next day.  A few weeks ago I forgot to pick up the mail at our rural box, it's secured so I waited until the next day.  But I had checked scans so I knew there was a book, but it was after 10 pm when I remembered and I wasn't walking down our long driveway in the rain and dark.

I wait about 7 to 10 days also if I have a DC, some people go to central mail boxes only about once a week and others are out of town.  Many people are appreciative of the PM though because they just forgot, it was picked up by someone else who forgot to give it to them, or was left by USPS at a door they generally don't use. (The last situation actually happened to me one time.)