Dear Lisa,
It seems to be that this man is wearing epaulets of the Colonel (with no stars), which was rather high military rank in Russian Empire mostly in later times, which meant 6th degree in the "Table of Ranks" and highest social status. There is a little chance, that he had a rank of General of Cavalry (similar star number), but I could hardly imagine it, judging by his appearance and Orders. However I may be wrong, because I'm not a big expert of definition Colonel/General epaulets. (Maybe Mike make some corrections;).
I can only envy you, having such a decorated ancestor! Almost all the Orders are with swords, that's mean he was a honored war veteran. I can clearly see St. Stanislaus with swords of the 1st degree, St. Anna of the 4th and 3th degree (or 2nd), both with swords and St. Vladimir of the 4th degree with swords. I can make a mistake of definition of St. Anna and St. Vladimir, but anyway I think I'm right in this case.
About the problem: "before 1881 or after", I think that I clearly see Nicholas II coronation medal (coronation was in 1896), so I think the photo was taken after 1881.
And about Regiment, difficult question. But I can make some thinking. If your great-grandmother was born in Pyatigorsk, then he definitely should have been served in some of the Caucasian region Cossack regiments (or even be commander of some of them). As far as I know, there were a big number of Terek Cossacks Regiments in this region. Here are the main: 1st Kizlyar-Grebensk of the the General Yermolov Regiment, 1st Gorski Mozdokskij of General Krukovsky Regiment, 1st Volgsky Regiment, 1st Sunzha-Vladikavkaz of General Sleptsov Regiment, 2nd Kizlyar-Grebensk Regiment, 2nd Volgsky Cossack regiment, 2nd Sunzha-Vladikavkaz Regiment. I can assume that he was served (or headed) 1st or 2nd Volgsky Cossack regiment, including judging by the letters on his epaulets.
Yes, it's possible to find out his name, but it requires more information and would take a plenty of time.