My crowned tooth fell out of my mouth with the post attached. Do I need to have the tooth root removed?

If that tooth really is solid with no decay present, it could be recemented. That would be the least expensive option. Most of the time however, when I examine a tooth like this, I find decay or a fracture or both. Patients are often unable to detect this in their own mouth and the tooth seems fine to them. If decay is present, the best case is a new post, core and crown. The worst case is extraction. An implant is much preferable to a crown on a short compromised root.

This would really depend on why the crown and post came out. Usually, it is due to poor condition of the tooth. In these instances the tooth has to be removed most of the time. If there was a cement failure and the tooth was fine than it may be possible to just re-cement. You would really need an evaluation and X-ray to determine the condition of the tooth. Good luck!