Yes, you can't register a company at your home address, but no one will ever deny you coverage if you do clerical work at home. At least every single insurance and rental I had in UK had a special provision for this - if you need to stay at home and work remotely, that's 100% covered. It just can't be your principal place of work.
Plenty of people register businesses at their home address. It entirely depends on the nature of the work as to whether it requires a change of use from the planners or invalidate insurance etc
You can easily get cover for working at home. But usually work is excluded from domestic policies; particularly third party liability.
Obviously the liabilities vary, if you're doing prototyping of fireworks in your garage then it's a little more risky than coding!
Insurance isn't always priced solely by risk; work-from-home insurance probably reduces claims for many people but that doesn't mean you can't charge more for it.
It would be interesting to know how occupier presence varies with claims made.
Contents & Home insurance - and I guess yes and no. Probably lowers the chance of theft claims, but increases slightly the chance of fire damage and accidental damage to contents. Don't have the actual data to back this up though.