A doorbell circuit for two or more doors will have a separate contact on the chimes for each button included. At the chimes, one wire from each button is spliced to the output wire on the transformer. The second wire is connected to one of the contact screws on the chimes.
At the switch, the other end of the black wire is connected to the bottom terminal on the switch and the black wire going to the new light is connected to the top terminal. The white wire from the receptacle is spliced to the white wire going to the light, it doesn’t connect to the switch in this diagram, but a new electronic switch will usually include a neutral wire connection. At the light the black wire connects to the hot terminal on the fixture and the white goes to the neutral terminal.
It’s not unusual for an old house to have an old doorbell lurking about somewhere that hasn’t worked in years. Fortunately, it’s likely a simple matter to get things back in working order. As illustrated in the above diagram, the parts of an old doorbell circuit include the bell, a button mounted at the door, and a battery somewhere in between the two.
This drawing illustrates the wiring for a split receptacle controlled by a switch. The tab connecting the hot terminals on the receptacle is removed and the source hot is connected to the bottom half. The source neutral is connected to one of the neutral terminals on the receptacle.