Yes, Russian diopside and chrome diopside are the same gemstone. Chrome diopside is a variety of diopside that is deep green in color due to its chromium content. It is often referred to as Russian diopside because it is primarily sourced from Russia.

chrome diopside and quartz are two different gemstones. Chrome diopside is sometime found along with quartz and hence the name.

Most reports of chrome diopside are found in Libby, Montana. Though there have been reports of finding these kinds of diopside in Canada, South Africa, Russia and Brazil.

Rarity is relative, and debateable. I think it's safe to say that diopside is not considered to be a rare mineral, though large, transparent green well-formed crystals of diopside are indeed rare, as are fine, large and clean faceted stones.

"Violane", the manganese rich violet to blue variety of diopside, is quite uncommon, found in Italy, Greece, and California, according to mindat.org.