Coolant gauge has stopped working

How long since a tune-up? Check applicable trouble codes. To check codes yourself, at the least, you need an obd2 code reader.

Check if the wire has not come loose from the sensor, if not, remove the wire from the sensor and hold it against any clean spot on the engine block whilst the ignition is on, (you need someone to be inside the car) if the needle travels all the way to the hot position, you will need to replace the sensor

you have probably been misinformed. The needle on the temperature gauge is controlled by the temperature SENDER unit ( not the temperature SENSOR unit) The sender unit is a single wire unit that is a resistance unit affected by heat of the coolant, The hotter the coolant the less resistance the higher the reading
The temperature sensor unit is a 2 wire unit that tells the ECM when to switch on the fan, when to go into limp mode if overheating and when to adjust the air/fuel mixture as the coolant warms up

the temp gauge works of a different sender to the coolant sensor. IT has 1 wire only and changes resistance as the water heats up there by moving the needle on the gauge up the scale Check that the wire is tight on the sender unit and if so then replace the sender unit.. The difficulty in re starting the motor may be because it is over heating but you would not know this because of the faulty sender unit.

Faulty coolant temperature sensor and a faulty heater control valve.

Its fine, temp needle going up shows that engine is warming up and as soon as the radiator fan starts, temp of coolant drop and this is shown by the temp gauge needle. In most cars (even mine) it stays in middle most of the time unless you are in a traffic with AC on, it will show up more temp. What you can do is get the radiator flushed and coolant changed. Also check the water pump. To do so when engine is cold (in first start in morning), open the radiator cap and ask someone to rev the engine. If the coolant level moves up and down then water pump is also fine.

if the temperature gauge never moves its probably just the coolant temperature sender. Replace that. but i honestly recommend buying a new temperature gauge with more accurate readings. they arent very pricey and they are easy to install

Locate the coolant temperature sensor it is usually located near where the upper radiator hose goes into the top of the engine drain the coolant from the petcock at the bottom of the radiator and unscrew the sensor to replace.

It probably needs a thermostat check coolant fluid level make sure it has enough coolant then replace thermostat make sure antifreeze has a 50% water 50% antifreeze mixture.