As a Russian citizen, am I allowed to fly an aircraft for private purposes in the airspace of the EU (e.g. exercise the privileges of my PPL, SPL, BPL or national pilot license issued to fly Annex I aircraft, such as microlights, etc.)?
No. According to Article 3d of Regulation (EU) No 833/2014, it is prohibited - among others - for any Russian registered aircraft, or for any non-Russian-registered aircraft which is owned or chartered, or otherwise controlled by any Russian natural or legal person, entity or body, to land in, take off from or overfly the territory of the Union.
Exercising private pilot privileges by Russian citizens, regardless of them holding another citizenship, falls under the prohibition for Russian natural persons to own, charter (any form of renting etc.) any aircraft in the airspace of the EU.