As Ethel says, in a rented property, the tenants are responsible for paying Council Tax, not the landlord.
The landlord will only have to pay business rates if he is operating a business from the premises. (Leasing the building to someone doesn't count as 'operating a business).
What I suspect has happened is that the landlord has simply given the address of the house as the registered office of his business. This is perfectly lawful as long as any documents served on the company will be received by the company secretary (which is probably your landlord anyway). Many major companies have their registered office at an address well away from where they do their business. (A single room can be the registered office of many thousands of companies). It is simply an address where , for example, somebody can send documents to the company. When they arrive at the address, they are legally deemed to have been delivered and 'served' on the company.
When I ran a limited company, the registerd office was my father's house. This didn't affect his Council Tax and it didn't render me liable to pay business rates for his address.
Your landlord's arrangement might be a little unusual but, based on what you've written, I can't see that there's anything illegal going on.
Chris