One must also consider that a significant portion of the Caucasus region, which includes Georgia, Armenia, and Azerbaijan, was under Persian rule until about two centuries ago. It was during that time that the region was annexed by Russia after winning a war against Persia.
No, not as it's written, Tamta. I believe this version of it is Georgian. It could have derived from the Persian name Tahmina. Tahmina is a very old Persian name of the wife of the famous Persian fictional hero Rustam (created by Ferdowsi, a Persian poet and the author of the Shahnameh, which is one of the world's longest epic poems created by a single poet and the greatest epic of Persian-speaking countries).