Hi Hans Peter,
Thanks for the update. Your original data showed about 9 bits of noise peak to peak and you should see something closer to 6 bits. A lot of times this can be an issue with board layout. How do you have your analog and digital grounds connected together? Can you send me pictures of your layout?
Before collecting data you should issue the SELFCAL command. This will help to determine where the true zero voltage level is within the data.
Regarding your last question, if the input is shorted, then the level should be zero volts relative to the reference. The measurement AIN+ to AIN- can be anywhere within the common mode input range. With the buffer off this is at any voltage between AVSS and AVDD. With the buffer on, it has the restriction I gave earlier. When the input is at mid-supply, then you can collect data with buffer on or with buffer off and compare the results.
Best regards,
Bob B