Brake lights don't go through the headlight switch- - - -no reason to. The light signal will either go directly from the brake light switch to the rear of the vehicle if it's not equipped with turn signals, or through the firewall to the steering column if there are turn signals. The same bulb filament is used for both turn signal and brake lights.
Power is supplied directly to the dome light from the fuse block and a second wire is routed to the headlight switch terminal that is grounded when the light switch knob is turned fully counterclockwise. The dome light socket does not have a ground.
The light switch should have a separate terminal for parking lights, so they don't illuminate when the headlights are on. If you want the park lights to burn all the time, just connect the park light wires to the tail light terminal.
Jerry
