Many cover one or both home health & skilled nursing. However, if covered your cost sharing will be vastly different between plans as is the qualifications for the services. Some will cover hospice but that coverage is less common.
No, health insurance of ANY kind, including HMO, does not cover assisted living or skilled nursing. You need long term care coverage for those.
Many of them WILL cover hospice care, though - but it's usually for a very short period of time, like two weeks or less.
Home Health Care isn't covered in a skilled nursing facility. You'd need a long term care policy to cover non-skilled services in a nursing home.