I presume you are flying Kenya Airways Premier World Business Class, in which case your bags should be checked through as your Kilimanjaro flight also has a KQ flight number.
As you are in transit you don't need a visa for Kenya as technically you do not enter the country whilst in transit. As you already have a Kenyan visa for your subsequent visit, it would cover you anyway if you did leave the airport.
I have never had a problem with bag transfer at Nairobi and I've been through well over twenty times. they will tell you whether your bags are checked through when you check in for your first flight, and you can check the luggage stubs to confirm.
As the Arrivals hall was destroyed in the fire they have temporary procedures in place, I last came through soon after the fire when they had various tents in place and departures and arrivals were scattered around various locations. Since the beginning of October all departures have returned to their pre-fire location in what used to be both the Departure and Arrival terminal, but is now just for departures. Arrivals are now in a temporary Arrivals terminal in the multi-storey car park across the open air car park from Departures.
When I came through they had a special arrangement for transit passengers, we were met by buses when we got off the plane and the first stop was transit passengers. Although the procedures changed in October, as the temporary Arrivals Terminal will not have any plane stands I would guess you will still be met by buses on the tarmac and as a transit passenger either taken directly to Departures or taken to Arrivals and then escorted to Departures.
As a business class passenger you should have access to the KQ Business Class Lounge if you have time spare