Yes, the same as in Ukraine, in Russia some categories of convicts may voluntary sign contract with army.
No.
If the crime is minor or moderate, not related to treason, drugs, a crime against the person (pedophiles, rapists, premeditated murderers), such people can sign a contract with the army after the investigation into their case is completed, after the charges are announced in court, the parties begin to debate.
The criterion for being allowed to sign a contract is that the criminal article implies imprisonment for 3 years or more.
Signing a contract occurs after a voluntary admission of guilt and does not exempt from liability - such a person is considered convicted, and after returning from the conflict zone, all the same restrictions are imposed on him as after release from prison - mandatory checks by the preventive service during the probationary period, restrictions on the rights to hold positions in government organizations, etc.
"During the period of military service, criminal proceedings against such persons will be suspended, and the preventive measure (recognizance not to leave, prohibition of certain actions, bail, house arrest, detention) will be cancelled." http://duma.gov.ru/news/60026/