I would also check the temp with a laser gun if you suspect the gauge.
I live in a hot place (around 30-40c in this time of year), fan kicks in exactly when half of the gauge. and when driving outside it can be a little bit below that but no quarter.
I use genuine and old stock parts, thermostat should be 87-92 and also the temp sensor on the radiator.
But, that doesn't seem related to your problem.
I would also check the fuel filter and to see that there is no blockage in that system (fuel pressure test).
There is also fuel regulator on the manifold, check it's vacuum pipe that is attached from it to the manifold.
Is the engine working ok on idle? if it is, try to remove each HT lead, might be also be a faulty coil unit or HT leads were connected the wrong way.
If the injector light still stays on, there is no really a way to check it but to replace parts or to visit a fiat garage that still owe the tools to connect the OBD port.
חג שמח ושנה טובה!
Thanks fot the new tips and suggestions - I will do them all and inform - didn't have much time last couple of days.
Fuel filter: I am a bit confused with this car: the only fuel filter I found (and replaced with a new one) is located on the right-hand side of the car, in the rear, above the rear wheel (under the fuel-entrance pipe to the tank, I guess). Is this the only filter, or am I missing here one?
Idle: I noticed it is working fine when I don't give gas - steady and quiet around 800 rpm - but the rev problem is happening also when standing: the engine is not reving up and "choking" and almost dying when trying to give gas and rev-up. so, when standing without giving gas is fine, but not when pressing gas.