Yes, you need to make your home insurer aware that you are running a business from your home. Even if the business is not inside your house but is being run from your premises such as garden, outbuildings or on your land. If you do not inform your home insurer and need to make a claim they may refuse to pay or void your home insurance policy.
This is incorrect, you are running the business from your home premises. This is known as a material fact and must be disclosed. If this is not disclosed the insurer can legally refuse to pay any claim.
No, Public Liability provides cover for third party property and third party injury.