There are two temperature sensors on the bike, the sender for the temperature guage is on the top left of the cylinder head, take the wire off and ground it - if the guage goes to full scale them the wiring & guage are OK. The sensor for the fan is in the back of the radiator, on the left side, pull the connectors and join them together to check the fan and it's wiring are working. You'll know it when the radiator fan is running - its quite noisy.
In practical terms the temperature guage on my Trident rarely comes off the bottom of the guage, except for when I'm in traffic. The fan wont come on until the temperature guage is above the halfway point.
Idling speed is adjusted easily enough, there should be a knob beneath the carbs, on the left side. Turn it anti-clockwise to lower the idling speed when the engine is properly warmed up. If you suspect the chokes, check that the chokes are fully closing when you move the lever back, I have had trouble with the choke cable getting dirty / stiff and not working properly.
(If you cant sort the Trident, I expect there will be a few cheap BMWs around your way sometime soon

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