Can I put negative voltage to the signal input of the ADS8317 A/D if I use a sufficiently large series resistor?
Those values (-0.3V and 20mA) are only to avoid damage to the device - they don't guarantee correct operation.  One side effect that can happen is that current gets injected into the substrate of the device and can cause all sorts of problems including causing other channels to give incorrect measurements. A sufficiently small current will probably be OK.

In the recommended operating conditions (page 3) they specify a worst case of -0.2V to avoid issues, they don't however mention the current.

If you want to measure system offset you need a "live" zero as you would get the same output from the A/D whether you had 0 or -1mV going into the A/D.

One solution I have used in a similar situation is to intentionally add an offset into the signal conditioning so that a zero input to the system gives something like 10-100mV into the A/D itself (depending upon the system).  You can then accurately measure the offset whether it goes positive or negative from the normal value.

To protect the A/D from negative voltages I would feed it from a rail to rail opamp (the same one I add the 10mV offset into).  The opamp then needs protecting with a shottky diode or similar..