Can the foreign citizen visit those cities which are not specified in his visa?

Yes, he can as his visa is issued to visit country, not any certain city.

Can the foreign citizen prolong the Russian visa and for what term?

According to the Russian Federal Law a visa can be prolonged by the authorized state body no more than to 10 day in unusual cases during the foreign citizen's stay in Russia and at presence of the motivated request and the documents confirming necessity of such prolongation.

Can the foreign citizen receive the Russian visa 7 months prior to his entrance date to the Russian Federation?

No, he cannot. According to the rules of Passport and Visa Department (UVIR) of the Moscow Municipal Department of Internal Affairs the foreign citizen can apply for his invitation letter not earlier than 45 days prior to the assumed entrance date.

Is it necessary for the foreign citizen to fill in a migration card when crossing the Russian border?

Yes, it is a necessary condition. A migration card is a document that contains the data of the foreign citizen, who visits Russia. A migration card allows to control over the foreign citizen's time stay in Russia. Also it is required for the visa registration. The registration mark is put down in the column " Marks about registration at the temporary residence " on an underside of a migration card.

Is it necessary for the foreign citizen to register his visa upon the arrival in Russia?

Yes, it is. It is an indispensable condition for the foreign citizen if he intends to stay in Russia more than 3 working days. If the foreign citizen has not registered his visa for any of several reasons, he can have problems at departure the Russian Federation. At the discretion of Customs service the certain sanctions can be applied to him, or it will be necessary for him to pay the penalty, or he can not be allowed out from the country.