The current zero test (29 pA) seems to be consistent with the bias measured in DCV mode (around 20 pA). The effect of shoring also is small, so likely no problem here.
The turn over test kind of indicates a problem, either with an offset or with INL errors. A few more points towards even smaller voltages (in the 10 V range) could help to distinguish offset and true INL a little better.
One source of nonlinearity are resistors, e.g. due to self heating. This effect should be smooth, mainly an U³ contribution. The main candidates are the resistor at the ADC input and the gain setting resistors in the 1 V range. Self heating takes some time and thus the 1 PLC mode might act different from a true 100 PLC mode. A comparison of 100 PLC mode an averaged 1 or 10 PLC mode might give some clues. However slow settling might have a similar effect - thermal effect can be one part of a slow settling contribution.
From Mickles description the internal cal for the 10 V 1 V step is done in 10 PLC mode. So it would be worth a test at 10 PLC.
A second source for INL is DA in the integration cap. In a multi-slope converter this is a rather odd jagged function with possible hundreds of points going up and down over the range. This is kind of difficult to measure and test, especially without PC connection. One could repeat the turn over test with a few (e.g. 5 points close together, like 1 V, 1.02 V, ...1.1 V). DA caused INL would give a more random up and down, while resistor caused INL would give a smooth curve. Depending on how the converter is made, there may be an additional smooth function INL caused by DA. Here positive and negative direction can be quite different as there is a constant current added at the input of the ADC.
The capacitor can effect different voltage different.
In theory there might be an error if the resistors for the current source change in ratio. I am not sure if and when the DMM checks those and measures them - it should be possible to use measured values, but I don't know if such a measurement is done, and if so when. It could be part of the ACAL procedure or a separate point.