For all your activities (except those related to powered vehicles like autos, airplanes, boats, etc.) your Homeowner's Policy's Comprehensive Liability provides coverage for any allegations of negligent acts. This usually includes incidental business activities that occur at your residence premises. (However, there is no coverage for "defective products" that you might sell or produce or for your activities off premesis.)
Umbrella Policies can be one of two types. In the first type, the Umbrella policy is excess to the underlying auto or HO policies. In the second type, the Umbrella can provide some coverage for liability not included in the underlying policies. Policyholders have to ask their agent which kind they might have. An example of possible coverage that might be included in a broader policy would be for serving on the board of a HO association, or other non-profit.
The underlying coverage must be maintained at the limit of liability required in the umbrella. If one of these has lapsed, say for missing a payment, then the Umbrella would have a deductible equal to that underlying limit before it would respond.
The challenge of our system of civil compensation is that justice often means that the person/entity who has funds/insurance ends up liable in order that an injured person can have medical or other compensation paid. This means that people with assets need higher limits of coverage than do the poor, who are dropped from suits when it is clear that they have no funds.