Are there types of household mold that are normal and safe?

Yes and No. Mold spores are found in the air most everywhere, especially in areas that are humid and warm. Some spores are very resistant to extremes of temperature and humidity so can survive in many environments. It is only when levels get high that we start to worry about the safety and health issues. Some types of mold can even be found on normal human skin.
Yes. There are only a few strains that are truly toxic to most people.There are specialty inspectors that can test for these strains.Some people develop allergic sensitivities to a variety of molds like you might to dog dander. This is a nuisance but not a danger to most.
Mols are normal. In most climates indoor mold exposure is expected and non-infectious to all but severely immunocompromised individuals. Many of these "normal" molds can give off spores that can induce allergic responses in atopic (allergic) individuals. A common mold, aspergilis can cause infectious (aspergilosis) or immunologic disease (allergic broncopulmonary aspergilosis - abpa).