Did Russia or the Soviet Union ever have their own experimental or military rocket planes (other than the Buran), or did their military merely use jets and other non-rocket aircraft? America flew those many experimental X-planes, did the Soviet Union have any similar planes? Afaik all MiGs are jets.
First, we should clarify the fact that a "rocket engine" is a variant of a jet engine. A rocket engine is usually understood as a jet engine that requires external supply of all components of combustion, including the oxidizer.
In a way, that makes things easier, and this is why rocket engines were commonly used at the early stages of jet propulsion development.
Early examples include rocket boosters for traditional piston-driven airplanes. This method is still in use to shorten the take-off distance (down to zero in some cases), but early on it was tried for "regular" emergency power and, of course, research. (Ram jets also had some limited use then).
The first Soviet 100% jet was BI-1 (БИ-1) in early 1942.
BI-1 had a rocket engine burning kerosene and nitric acid. Even though the airplane was designated as "fighter" and had a space provision for guns, it was really a research ariplane, with the maximum time under power of only a few minutes (it never reached even two minutes in real tests).
The first one was corroded away by the acid. The third one (BI-3) was lost, and the test pilot Bakhchivandzhi died, presumably due to Mach tuck, which was not understood then.
In the later times, during the "true" jet era, there was much less cooperation between the aviation and space industries than in the West. This is one of the reasons why "rocket planes" were a rarity in the USSR, even for research.