I have tried with both a GY-302 and a GY-30 and both don't work although they show different problems (mentioned on my OP). Both the sensors work well when connected to the pi.

I don't have a drawing of the circuit but I can show you a picture, if it helps.

The diagram is essential like this one but without the temperature sensor and I am currently using an FTDI board to provide the power, instead of a battery.

The SCL is connected to A5 and the SDA to A4 of the mini pro.

I can also do a drawing, if further info is required but, as I said, it is like the one I linked above.