Can I take out a life insurance policy and make the Russian government the beneficiary instead of my family?

Sure.  You'll have to name the country, though, as the beneficiary.

Only if your spouse agrees to it, at least in my state.  In a Will, yes you can.  You can leave your assets to whomever you want.  Just be sure you have an experienced lawyer draft the documents, so that your family cannot dispute/challenge your Will.

Where are you located?

Most locations outside of Russia the insurance company will probably refuse to issue that policy.

Also, regardless of location many insurance companies require that if you are married your spouse must be the primary beneficiary and/or they will require your spouse to sign a waiver if you want to take out a policy with anyone other than them as the primary.

It's not a legit business or corporation. so no.