So the barrel is forced/accelerated backwards trough the coil, closes the breech and hits the valve in the back of the loading-mech?
Do you drill out this piece from the top after soldering it together?
Yes, and you better be careful while doing it!
Is there a spring in the back of the valve (in the elbow) keeping it closed, or does it close from pressure?
Why is the gun still shooting a few shots even though you already removed your finger from the trigger? Is it because the motor is still turning from inertia?
Yes it is due to the rotating mass of the motor, aka inertia.
If that is so and if you don't like that, you can maybe avoid that by having the solenoid in the same circuit as the motor, so the solenoid would not be able to work after you stopped pushing the trigger.
Just a thought.
That would, in this specific design, jam the BB feeder. If you feel the need to remedy this non-distinctive ending you can modify the trigger system so that it short circuits the motor when the trigger is released to momentarily stop the motors rotation. I like to think about it in terms of that there is room for improvement and customization!
I wish you good luck with the sale of your plans!