Most duplexes & triplexes allow #2 & 3. The owner occupies one unit and has full rights & responsibilities over the property: they can paint it, landscape it, put in solar panels, etc. Then they rent out the other unit(s) as a normal landlord/tenant relationship. It's extra income for them, and it's a normal apartment for the tenant (except often a bit nicer because many duplexes have yards and the owner has a stronger incentive to fix things quickly).

Townhomes/condos with an HOA are a different beast. That's a function of the ownership structure (SFH vs. condo vs. TIC vs. apartment) rather than the physical dwelling, though. At least in the Bay Area, it's not uncommon to find SFHs that have been converted into multi-family properties with the addition of an in-law unit or ADU. The vast majority of these use a traditional landlord/tenant relationship.