Tromso-Nordkapp-Murmansk-Rovaniemi road trip

Oh, and also, is it going to be a problem taking a rental car from Norway into Russia?

Actually I've been planning a similar trip, as I live near Tromso, I want to hitch hike from Tromso to Murmansk this summer. I want to take a bus from Kirkenes (Norwegian border town) to Murmansk, then a train from Murmansk to St. Petersburg.

First things first;
Rovaniemi is in northern Finland.
Riddu Riddu is a indigenous peoples festival held in Manndalen, Norway, a valley a few hours drive north of Tromso.

Camping is allowed everywhere in Norway, just bring a tent and stove. You don't have to pay for a campground. You can camp for free at that festival as well I believe, just reserve your concert/festival tickets in advance online.
I don't recommend camping in Russia, far to dangerous. Northern Norway however is completely safe.

I think Kirkenes-Murmansk is about 5 hours.
Tromso to Kirkenes or Nordkapp, I don't know, over 10 hours at least.

Why not drive to Lofoten, Senja, Hinnoya, Vesteralen, Kautokeino or Karasjok instead?

Russia will be a huge hassle with the visas and car permit, so you should research this in advance. The visa is expensive, requires an invitation etc. Hotels in Murmansk would be expensive.
I love Russia though, I actually live here.

Hmmm, instead of driving into Murmansk, would it be easier to take the bus from Kirkenes?

Also, if you had about two weeks, what would you do and see in Arctic Norway, Sweden, Finland and Russia's Kola peninsula?

The buses from Kirkenes are organised at the embassy there or something, I haven't been there nor Murmansk yet, only done the research. It's quite expensive, but Norwegian buses are more expensive. And you'd have to pay for parking while you were in Murmansk.

I haven't been to the Kola peninsula, only St. Petersburg.

If I only had two weeks I would start the trip in Stockholm, take the train to Narvik, it's about 20 hours I believe, a very scenic ride! The same price as flying but with a better view. You could fly back to Stockholm via Oslo.
I'd definitely do the Riddu festival. I'd skip Finland and Russia though, on this short trip.
I'd go for Arctic Norway, as it has fjords, steep cliffs etc. I don't know about the itinerary, but be sure to include Tromso, Vesteraalen/Lofoten, some Sami culture etc.

My two cents - No major car rental company will allow to take the car into Russia. Neither into any baltic and Eastern European country.

RE Driving times: Depends on how you drive. If you drive fast (which is not easy to do in Norway), you can on average cover ~700km per day. with rest and everything, provided that you enjoy driving.