I don't think the missing-characters business you're encountering is related to BibTeX, at least not directly. I downloaded the files you posted off-site and have had a look. I'm very sorry my Russian -- and even my ability to read text that uses Cyrillic letters -- isn't all that good. The following is therefore mainly going to be a set of suggestions, which you may want to treat as no more than a starting point for further investigations.
Since your files are utf8-encoded, I strongly recommend you use a TeX engine that's fully utf8-enabled. I suggest you use LuaLaTeX. (I assume you've been using pdfLaTeX so far. Is that correct?) If you choose this route, you should comment out (or simply delete) the instructions \usepackage[T2A]{fontenc} and \usepackage[utf8]{inputenc} from the preamble of the main tex file. In their place, (a) insert the instruction \usepackage{fontspec} and (b) choose a font (via a \setmainfont instruction) that's on your system and has excellent support for Cyrillic characters. On my system (MacOSX 10.11.6; MacTeX2016), I found "Garamond Premier Pro"; I thus wrote \setmainfont{Garamond Premier Pro} in the preamble. I trust you'll have several good fonts to choose from on your system.
Run LuaLaTeX once.
Next, instead of running "basic" BibTeX, which can probably handle only ASCII-encoded material without generating meaningless error messages, consider running bibtexu, which is utf8-enabled. You will probably need to open a command window, switch to the directory that contains your main tex file (SampleFile.tex, say), and type
bibtexu SampleFile
Ignore any cryptic error messages.
Then, run LuaLaTeX twice more to generate both the formatted bibliography and citation call-outs.
I get the following output from this exercise; is this roughly what it should look like?

\documentclass{article}
% I've shrunk your tex file down to the absolute minimum...
\usepackage{fontspec}
\setmainfont{Garamond Premier Pro}
\usepackage[russian]{babel}
\usepackage[square,numbers,sort&compress]{natbib}
\bibliographystyle{plainnat-re}
\begin{document}
Citations: \citep{Ajn1962, AbrAru1963, Bab1984, AlbN1972}
\bibliography{sample_bib}
\end{document}