NM -- Keep in mind that Kiev is in Ukraine, not Russia, thus you need to consult with the Ukrainian rules. According to the Ukrainian Consulate General in NY, "registration with local authorities of foreign citizens temporary visiting Ukraine and their passports is no longer required as of July 1, 2001." You still need a visa though.

Your options for transport from KBP to Kiev will depend on how much luggage you have. One economical option is to take an express bus (reasonably comfortable) from KBP to the city center, then hail a cab (would be my personal choice) or take a subway.

Hope it helps!