New language request

To avoid much discussion, are we able to make translation in our own server using some MediaWiki extension or something else?

Yes, add a file called 'MessagesXxx.php' where 'Xxx' is the language code in the folder where 'MessagesEn.php' is. You can copy from MessagesRu.php for example. Then add an entry for 'xxx' in Names.php to make the code active.

And is it possible to automate this with the Translate extension, based on the Russian original messages from official MediaWiki repository? At the same time use the same extension Translate to translate some of the required extensions of MediaWiki on the Siberian language (in our own server)? What are the configuration settings for the Translate extension necessary for this?

See Translating:Documentation.

Yes, I already read this page. For translation to Siberian with this extension should only add a file called 'MessagesSib.php' and «'sib' => 'Сибирской'» in Names.php and this language will be available via extension?

And how to define a language — the source of translation?