Does that black elbow unscrew from the sink fitting leaving you a 1 1/2 thread to accept the 2 bowl kit? Doesn't look like it but your picture doesn't show that bit very well.
Trap comes with 2 little blue plugs which blank off these connections and I always leave them in the sink unit so that if the appliance has to be removed, sink can continue to be used.
Otherwise, wine bottle cork or remove nozzle and stuff with tissue, bit of rag etc.
Replace the basket strainer waste as said.
Sink has an overflow so bit of tape over that and plug washing machine /dishwasher connections. Now fill sink with hot water and plunge.
Mixer taps (including kitchen) would only be a problem when used. A shower or blending valve would not need to be in use. Do you have a shower with tank fed hot and mains cold?
It may be that when heating is running aavs are at negative pressure so allowing air to be drawn in.
With heating off and after the noise has stopped, tighten the dust caps on the aavs.
If this cures it then replace aavs with manual vents
With heating only the boiler temperature is controlled by o/s sensor.
If hw is then required, boiler would not give enough heat to get the cylinder hot.
Hw on will override o/s sensor till cylinder is satisfied then hw off will hand back control to o/s sensor