Ok, I've had my time with the sele on my 147.
Here's some of what I’ve learned.
If the engine is off, but key is at "Mar" you should be able to select all 5 (+1) gears. With the engine running and the car stationary, only 1st, 2nd and Rev are available. Neutral is available in all these states, but not when moving above a speed threshold.
The pump only runs if the pressure in the accumulator is below a measured line. (Unless told to by a diagnostic program, EG. AlfaDiag)
While the car stands, the pressure slowly releases and the fluid moves back to the reservoir. When you open the door the system activates. If the pressure is low then it primes, to build pressure in the actuator. If the pressure is high enough it won't prime.
So opening and closing the door won't make it prime every time. The computers also "Shut Down" if nothing happens for a while. (Like when packing the car for a trip or at a picnic.)
Your next step is to open the window (With the engine off), shift into first, then try for second. Listen to any noises it makes. You are listening for the distinctive “chunk, clunk” of a attempted gear change.
That's a big clue as to where the problem lies. If it's not making any noise but is priming then you need to look on the control side.
If it's not priming then that's where you need to look further. Like Pump/ relays / fuses.
Else it's heading back for Actuator or physical damage. A shot clutch / damaged gearbox / failed actuator.
How hot is it in the car? Mine has a weird issue with a hot passenger cabin heating the ECU and causing all sorts of mischief.
Report back with what you find.
Oh and as far as I’m concerned, the only what to properly check a battery is to put it to a load test. The nearest battery center / dealer should be able to test it for you. A pure voltage test, checks the current charge state of battery, not the condition of the battery. IE, a battery can be reading fine on the voltage, but the second you actually load it then it collapses. A load test measures the voltage profile as it loads the battery.