Mandarin is not my area of expertise, but I would not be surprised to learn it had been influenced by the neighboring languages. Every language is subject to influence by its neighbors--it's more or less an inescapable fact of language contact. One way languages influence each other through contact is lexical; languages may borrow lexical items (words or phrases) from their neighbors. English, for example, has borrowed thousands of French words, such as 'court,' and 'beef,' going back to the middle ages. Languages are also frequently subject to phonological influence from neighbors--they can adopt speech sounds from each other, such as certain types of consonants. Some African languages famously include "clicks," an unusual type of consonant that includes making a loud clicking or popping sound with the tongue or lips. Languages of the Khoisan groups have inherited clicks from the predecessors, but languages of the Bantu groups didn't originally include these sounds. Modern Bantu languages, however, include clicks because they have borrowed them from neighboring Khoisan languages.
These are just a few of the ways in which languages can influence and be influenced by their neighbors. Interestingly, the languages you named all belong to different proto-families, Russian being Indo-European, Turkish being Uralic, and Mandarin being Sino-Tibetan. Nevertheless, these are all thriving, widespread, and geographically positioned to exert influence over each other in phonological, lexical, or grammatical ways. Sorry I couldn't provide more specific information about the influence of Russian and Turkish over Mandarin.
