Where and how to register for free brand and logo, or even to get a patent on the name?

Free - no way.
It costs about 30 000 roubles. Takes about a year.

Can score (almost free) domain.
If the trade mark is registered by someone else later, your domain registration and scope of activities are different - we can still wrestle in court.

The concept of the brand in the Russian legislation there is, it is not registered... is a trademark, it is registered, shall be issued a certificate for certain types of activities in which it will be used. The brand is a broader concept than trademark...

The approximate cost of registration, announced to you. Before you apply for the registration of TK (directly or through third parties) make the information search is similar to TK, or may be denied registration, will lose the stamp duty (this will be a bigger part of your costs).

Information search can be done independently, but a fee, read more here.

Domain could not be checked, availability of the domain, or matches its name with any TK does not mean that you will not be able to register their TK, because the value in the first place, has a view activity (nice classification in terms of Rospatent), which was specific to TK.