barthulsen:
Since I've read that the other analog ports are unusable during free running mode on one of them, I'd like to ask the following question: Do you expect that this will cause any problems? Or can I just simply turn the continuous sampling on and off in the readData function to use I2C on other ports?
You can't use analogRead() to read another Analog input pin while doing continuous conversion, because analogRead() will reconfigure the ADC registers such that the continuous conversion stops. I don't think this applies to using one of those pins (A4, A5 in this case) for alternate functions, such as GPIO or I2C. Thus I don't expect there's any issue with I2C to your display while the ADC is running, but it's not something I've tried.