We'd just moved when you posted, completely missed it...
No, no government registration needed. You CAN register with the Home Office if you want, but you don't need to.
All you need is your NI number.
We didn't have much difficulty in getting a flat, only paid one month's deposit as well (with an agency). I would stress the Scandinavia bit when looking, Scandinavia is well known for being 'upscale' with housing, so I suppose landlords are more comfortable renting to Scandis.
Or so it seems.