Hello Lagartija Mike!
In George Nafziger’s book Napoleon’s Invasion of Russia he shows an Opolochenie 1st Cavalry Reg., but it only contains 126 men and was probably just a support unit. He also refers to the Cossacks raised in Tver on Sept. 12, 1812 as Opolochenie. This is probably just an editing error. Regardless these are only about 700 men. I’m not aware of any other reference to them having cavalry. So I think it’s a pretty safe bet to assume that they had no battle cavalry.
He shows that the Opolochenie were used to replace losses in the regular regiments and were also assigned as whole units to the various Divisions. It seems that unless they were stationed for home defence they were generally allocated to the main armies.
In general the Cossacks were either within their own formations or brigaded with other cavalry forces. As such it would seem that the Opolochenie interaction with Cossacks would not be very common.