The oscillation at 2.5V input without the .01uF cap is difficult for me to explain assuming all the usual steps of proper bypassing are in place, short leads, and good routing. Hmmmm.
I'm puzzled for the need of the .01uF capacitor at the output of the inamp if the microcontroller you are using is the Freescale model in the link you gave previously. In the datasheet for that micro, page 37, it specifically states (for reason unknown to me) that the R/C time constant of the output resistance of the source (80 ohms at 300KHz for the inamp) and the output capacitance of the source (added .01uF cap) should be less than 1nS, which it is NOT. Based on that, if the .01uF cap is for filtering the 5mV of noise you mentioned at the input of the inamp, you should do the filtering between the op amp buffer and the input of the inamp (as a described in my last post) instead of the output of the inamp, and/or before the input to the op amp buffer, and forgo the .01uF at the output of the inamp. Then, it's not clear if the note in the datasheet means Rs/Cs as a division operation, or the "time constant" which is Rs*Cs.
Something curious is that the datasheet for the micro gives an input resistance of 2Kohm typical, but in figure 10 that resistance is shown in series with the input of an "ADC SAR Engiine", the input current of which is not specified.
I'm interested in how your experiments work out, in any case.