Great questions, Midas.
My wife and I bought our duplex when we didn't have kids. The location is decent, but the layout is what sold us. It's on a bus line, first floor and basement are one unit, second floor and loft are the second unit. It also has separate laundry facilities.
Each duplex is different. Ours has one front door that opens to a foyer area with a stair case to upstairs. So, in this case, yes there are two locked doors one to each unit. When we had kids though, we decided to buy another single family place and rent out both halves of the duplex. In our market at the time, this made us more money (both cash flow and depreciation wise) than what we pay per month for the single family place. Other duplexes have separate outside entrances (two front doors for example). Some of these may or may not have a shared staircase in the rear for access to the basement.
Insurance on the property covers the structure and your belongings. It DOES NOT cover the renters' possessions. I encourage, but do not require, my tenants to purchase renter's insurance.
Utilities depend on the unit too. Our duplex has shared water and trash (which is owner paid) and separate electric, gas, phone, cable, etc. billing. I had friends who lived in a place where everything except phone, internet, cable was covered by the landlord. They paid a premium for that place though (I think they could have done better paying on their own). There is also no incentive to conserve utilities, so there's that side of it too.
Regarding existing tenants, yes you typically inherit the leases if the unit is sold. However, there isn't often a provision that only one party can terminate the lease (i.e. not renew). You could also raise the rent and see what happens. Maybe you come out ahead renting another place and charging a bit more at your duplex. If you do end a lease, be careful not to do so in a way that may be considered discriminatory particularly if that unit's tenants could be considered a member of a protected class in regards to race, religion, gender, sexual orientation, etc.