Ok, thank you all for continuous support!
Just received my new sensors: a separate gyro and a separate accel (my current sensor as you see on photos, has both of them combined on one board).
I can't rely on my old sensor, not really sure about its health, she is too old!
With new ones I'd first build a new chassis, with the hope that it gives me room to easily apply changes as you suggest and instruct. This will take probably around 2 weeks and I'd be back here with a new bot which (hopefully) also helps to answer the questions you kindly put (how goes the output of gyro if you do this/that?) easy and fast.
Please, if you have suggestions for the new chassis I'd build, proportionality info., etc. let me know. What I know for now is that, I'd for instance build the vertical axis in around 70cm height, so that to have it falling not too fast, and keep the base in a way I can add and remove weight to easily change the moment of inertia. Any other suggestions will be very welcome, I will apply them in 3D modeling of chassis in software and finally cut with CNC.
Thank you again,