More below pertaining to existing flags of these units -
39th Tennessee Infantry - as I have stated a couple times here, no flag exists today.
Napier's Bttn. Cavalry (became 10th TN Cavalry) - no flag exists
10th Tenn Cavalry - no flag exists
16th Tenn Cav - no flag exists
1st Tenn Cav - there were two 1st Tennessee Cavalry regiments - Carter's and Rogers'.
No flag exists for Carter's and they lost their flag at Dutton Hill, KY (just north of Somerset) in March 1863. They had received it in February at Rogersville, TN by the local ladies. One Federal account of Dutton's Hill described their flag as, "the first sight most of us had had of the starred cross," which means it was an ANV style flag. The flag is missing today.
Rogers' 1st Tennessee Cavalry has an existing flag (probable ID) in the Tennessee State Museum (collection No. 73.10). Like most Western Theater CS cavalry regiments until October 1863, this was a First National. You can contact the museum for details and an image. The flag at TSM was taken at Roger's Gap in March 1862. The regiment also lost a flag at Cumberland Gap in August 1862 that is missing today.
As you can see there are a lot of missing CS flags today - hundreds in fact when you count 1861 presentation colors. I also have seven four drawer file cabinets loaded with files on CS flags, existing and missing, and there's just no way to make an index of what I have.