Hello and welcome
Interesting symptoms. The not-being-able-to-move part with clutch engaging and feeling as though the brakes are on, and engine stalling, sounds as though there are two gears selected at once. I wonder if the finger bolt has worked loose - the one that connects the actuator to the gear selectors. By removing the whole actuator from the gearbox and refitting, you could potentially correct this problem. But it would be nice to have a little more supporting information first.
The pump running all the time is perhaps because it's trying to select gears all the time. What exactly happens if you switch on the ignition without the engine running, and then use the lever to select each gear in turn? Does the LCD show each gear number in turn and does the Selespeed make the appropriate brief noises? Or is the LCD stuck on one gear?
Of course, it would help to use Alfadiag (www.alfadiag.net
) software which costs 60 Euro, and a KKL-type OBD cable from eBay, about 10 pounds, as we are always suggesting on this forum. FiatECUScan is another option, and free. Then you can see whether the system pressure is within acceptable limits (apparently the pressure reading is wrong in FiatECUScan, but you should still be able to see whether it is fairly constant). You can try an EOL calibration, which is where the system calibrates its gearchange position sensors according to where the actuators place the selectors. There is even a routine that diagnoses the loosening of the finger screw!
The accumulator (the big black sphere) could be faulty - in that case the pump would only run for a couple of seconds when you open the driver's door. How long does yours run for when you open the driver's door after the car has been sitting untouched for a few hours? The accumulator is a gas-charged sphere that the pump pressurises fluid against, a bit like an old Hydragas displacer unit in Allegro suspension, if that rings any bells with you
If the gas is lost, the pump builds up pressure immediately, but there is then no reserve of pressure to operate the actuators properly, so the pump has to run again while the actuators are operating. The accumulator is available as a FIAT replacement part for the Stilo Abarth/Prestigio models with Selespeed - do a search on this forum. But you definitely don't want to go down this path until you have connected the car to diagnosis software, to work out whether the system pressure is continuous and correct.