speak from my heart try to split your question in two.
If you want to find out russian words borrowed from the native languages of people of former Soviet Union then wiktionary has a category for this: Слова по происхождению (Words by origin). For example subcategory of words of turkic origin:
- Слова тюркского происхождения (Words of turkic origin)
Some of them are commonly used:
Also, thorn already gave a couple examples of Georgian words.
The second part is about republics.
All of the examples above are the results of a long-term influence of neighbouring turkic folks in VII-XIII centuries, but it's nearly impossible to tell exactly how much are they related to the modern republics of former Soviet Union. Words are borrowed from languages, not officially loaned from foreign countries %)
Usually loanwords came with the corresponding concepts. Soviet Union was founded less than a century ago. When the Soviet Union was established most of the "common" words were already present in the native languages. AFAIK, most of the words borrowed from russian language during the soviet era were neologisms, at least from the point of view of the borrowing language. The same goes for russian words borrowed during Soviet period.
There are proper names for territory-specific types of food (
бешбармак), musical terminology (
акын), geographic and ethnic names (
кишлак), etc. I believe that there are plenty of russian native-speakers who understand what does these words mean. But I also believe that most of them understand what are the origins of the word.
Comments to your update: *IMHO*
кочерга (poker) - etimology is unclear. Wiktionary links to Acharian's deduction to Armenian. But Vasmer's etymological dictionary derives it from Ukrainian/Polish or Turkic (thanks to thorn for remark)
мечеть (mosque) - both turkmen and russian words were originated from arabic مسجد (mæsdʒɪd)
I would separate well known proper names:
- Айвазовский (artist)
- Петросян - Comedian Yevgeniy Petrosyan, is sometimes used as a common name for not very funny comedian last years (mostly on the web). Generally Петросян is a quite popular Armenian last name. Consider, for example chess champion Tigran Petrosyan.
чача - food and drinks
тенге - currencies
- Ингушетия, Осетия, Севан, Сочи, Тбилиси, Цхинвал, Цхинвали - toponims
хачик - ethnically specific terms (some of them are offensive)