Rada is a Slavic name, so it is Russian as much as it is Bulgarian, Serbian or Croatian.
On the question whether many Bulgarians have Russian names: Bulgarians and Russians have a lot of common names, due to their common Slavic heritage (you can put it as Bulgarians use Russian names or Russians use Bulgarian names - both are right :). In addition, there are some Bulgarians named Stalin/Stalinka, Lenin/Leninka, and some are using diminutives that are more common for Russia than for Bulgaria, e.g.
Masha and Dasha for
Maria and
Teodora (the Bulgarian ones would be
Mara and
Dora). Those people are all in their 50's - 60's and are remnants of the ideology that preached that the Soviet Union is the best place on Earth and Bulgaria should become the 16th republic (luckily they did not want us!).