Is it possible to take your child out of the school system to homeschool them in the middle of a school semester?

Oh, if you're removing a child from pubic school, then it matters not a jot when you do it -- because you're opting out. On the other hand, if you're going to transfer a homeschooled child to public school, it's very important to do it at the beginning of a term, ideally at the beginning of a school year.

Yes, or at least with K12 (the school).

You can do it whenever you want but I would recommend being prepared with the appropriate learning materials. No need to wait a few weeks for books and whatnot to arrive. I was homeschooled starting in the beginning 4th grade but there were other homeschoolers in the area that were pulled mid-term because of issues with the school administration - mostly religious reasons.

Yes. You get yourself buzzed into school while smiling at the security camera, wave at the nice ladies at the front desk, head to your child's classroom, wait politely to get the teacher's attention, say, "Come on kiddo, we're going home," stop by the store for some fresh fruit and a walk in the park (perhaps while discussing photosynthesis and why the sky is blue), go home and write a letter to the principal, light a candle and celebrate your child's emancipation from the Industrial Revolution. Sorry, I think I went a little overboard there!