Will a temporal CT scan show tmj?

Possible. It's in the area, so it is possible to see TMJ, but probably not the right parameters, so unlikely radiologist will make a diagnosis without a formal CT/MRI of TMJ.
It should. If not done yet, you can ask the ct technologist to be sure the correct study is about to be performed, but it should show the TMJ adequately.
Yes, it should. Yes, it will. If you are having the scan to evaluate for migraine headaches and/or other pain causing disorders, the doctor who ordered the scan will typically request a report of findings. If not, your can request to have your TMJ evaluated. If you have other questions about tmj, feel free to email or message me. Best of luck!
Yes. Ct of temporal area of skull shows mastoids, ear contents, and tempomandibular joint. Gross bony abnormalities, subluxation, and arthritic changes (joint narrowing)in the joint can be detected. Sometimes images are done in open and closed mouth positions.