Welcome to Photrio,
@Stecco!
That surprises me greatly. Given the state that connector is in, I don't see how it would work reliably, if at all. Such FPC connectors rely on the pressure of the top part to press the FPC into the contacts below. Given that that part is gone, and glue is likely to have crept in between some of the contacts, I don't believe that this connection is very reliable. I wouldn't be surprised if it seems to work OK as long as you apply pressure to it (e.g. with your multimeter probe), but the contact breaks as soon as you release pressure. Also, under the mechanical stresses of normal use, this connection will likely not hold up.
The best solution would be to replace the connector with a new one; it's often possible to find a suitable part for replacement. Measure the pitch between the contacts and their number and find a new one.
Yes, and yes. It's quite possible that this dodgy connector plays a role in the problem, and at the same time, the question arises how this connector became damaged in the first place. This sort of damage makes me think it's due to someone trying to release the cable without knowing how to, and forcing the connector in the process. This in turn begs the question why they were working on the lens, and the reason could very well be the problem you spotted initially (no camera-lens communication).
I'm not too optimistic of actually finding the problem due to the lack of good documentation of the electronics and the difficulty of making a useful test setup.