Thanks a lot for your info, it's interesting. So, it seems that it's enough to have a registration stamp on your migration card to buy a SIM card, isn't it? It's not that bad, I mean they do not ask for a long term residence permit, at least. What's strange about Полиция Нового Орлеана? Here passports are being issued by милиция (i.e. police), so Russians think it's police who issue passports everywhere. Here it's common to use the passport issuing date and issuer's info to identify a person as well as other personal data.
MegaFon currently sell recharge cards valid for all of their branches, which are called 'Единая карта оплаты' and usually look as a green colored card with big white digits showing the card value in rubles on it's face. I haven't seen regional recharge cards (valid for the only branch) for long enough.
Accordingly to the
MegaFon Moscow site their voicemail should be activated by a call to *105*4*3*1# (Лайт tariffs) or *105*3*1*1# (contract tariffs). Had you tried such the combination? Also, it seems you have to set up call redirection (when not reachable or may be when busy etc.) to the voice mail number (+79262000222) yourself. Also, first time you call the voice mail number you have to specify your password and record a voice phrase. There are some other complicated options. Also, their voicemail is not free (monthly fee on some tariffs and 30..70 rubles just for switching it off). So, it's not that easy to get it working for a foreigner.
And, finally, visitors to Russia should keep in mind that CPP (Calling Party Pays) is now working here (since 1st of June). So, any calls to Russian SIMs (including calls from abroad) are free for their owners.