No, there is no train connection. No planes either. The bus from Kandalaksha to Rovaniemi, Finland (via Salla and Kemijärvi) runs twice a week. Lots of offers for Finland and Sweden from private car owners in Kandalaksha. The usual cost is around $30 one way to Salla. Transfer-tour (http://transfer-tour.ru/, link down as of April 1 2015)regularly sends minibuses back and forth. All the services cater to Russian tourists, with very few English speakers and no info in English. Buses run regularly form Murmansk to Kirkenes (Norway) and Ivalo (Finland).
..This is a preliminary review. More info is coming up. Talk to me if you need an individual trip and if my truck won’t scare you off. Think of $1/km as the average cost of hiring me.
..Alexandra recently went to Finland. An overnight trip cost her 4000 roubles ($120-something) including both trasnportation and accommodation but that was a bit of good luck.