Does/did Russia or the Soviet Union ever fly rocket planes?
Yes
First, we should clarify the fact that a &quot;rocket engine&quot; 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 &quot;regular&quot; 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.

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BI-1 had a rocket engine burning kerosene and nitric acid. Even though the airplane was designated as &quot;fighter&quot; 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 &quot;true&quot; jet era, there was much less cooperation between the aviation and space industries than in the West. This is one of the reasons why &quot;rocket planes&quot; were a rarity in the USSR, even for research.