As far as I know this tool, Fantastico scripts and other software are included almost in all hosting packages! I know exactly that this tool supports English language. If for example my website is written in Russian, will be the Russian requests under which my website is found in Google displayed? Thanks in advance!

Yes, awstats supports russian.

From the awstats FAQ:
http://awstats.sourceforge.net/docs/..._faq.html#LANG

Albanian=al, Bosnian=ba, Bulgarian=bg, Catalan=ca, Chinese (Taiwan)=tw, Chinese (Simpliefied)=cn, Croatian=hr, Czech=cz, Danish=dk, Dutch=nl, English=en, Estonian=et, Euskara=eu, Finnish=fi, French=fr, Galician=gl, German=de, Greek=gr, Hebrew=he, Hungarian=hu, Icelandic=is, Indonesian=id, Italian=it, Japanese=jp, Korean=ko, Latvian=lv, Norwegian (Nynorsk)=nn, Norwegian (Bokmal)=nb, Polish=pl, Portuguese=pt, Portuguese (Brazilian)=br, Romanian=ro, Russian=ru, Serbian=sr, Slovak=sk, Solvenian=si, Spanish=es, Swedish=se, Thai=th, Turkish=tr, Ukrainian=ua, Welsh=cy

As far as I know it does