State Farm's umbrella policy will cover your personal liability associated with the house, even though they didn't write the homeowner's policy for it, provided the H.O. policy meets the underlying coverage requirements that they (S.F.) have in place. I believe adding that liability coverage for the house will increase your umbrella policy premium.
Essentially, the umbrella policy will pay off on any personal liability that is beyond what is met by the base coverage(s) in the H.O. policy. Hence S.F.'s requirement that their underlying coverage parameters be met. So you'd have to check with your S.F. agent and see if the ASI policy meets their parameters.
Incidentally, it's not required to have your umbrella policy cover the house. In that instance, anything you could be held liable for from a personal standpoint that would be associated with the house would not be covered by S.F.'s umbrella policy. Same thing with your golf cart - if you didn't want it to be under the umbrella policy, then you can get whatever limits and types of insurance coverage you'd like.....but that insurance coverage would be the extent of the liability coverage you'd have related to the cart.
Hope that makes sense!!
Bill