There are imported and escaped rabbits in a few places in the south like around Helsinki. But not generally no, it is not a native. We had a kind of rabbit problem but a virus decimated the population 2 years ago and it has not gone up yet.
The bigger birds of prey and foxes and stray cats are having a feast with them. I don't think they would be able to spread around much more, the winter is also the big limiting factor.
Yeah I prefer the large scissor type traps for moles.
Connibears tend to be larger than a mole trap needs to be, they only need to open as wide as a mole run.
Otherwise fen traps or live catch cage traps are my go to.