Would using the reference voltage of an ADC as a VOLTAGE SOURCE affect the ADC or the stability of the circuit

It certainly looks feasible, but first make sure you NEED to do it. The 24 bit ADC gives you an awful lot of dynamic range that you don't need, and plenty of bits to spare. You often use them to sample small signals without worrying about preamplification.

So, work out your noise budget, figure out how many bits you need, and do what you need to get them. Try to remember that an engineer who consistently exceeds specs instead of meeting them is wasting time and money.

• The values of the resistors are fine, but I would suggest you use two 100k resistors to maintain the voltage around the center of the range. It's safer... And put in a resistor in series with the output of the transformer. Be sure to check if your transformer has no leakage resistance/current. The center point shouldn't change DC voltage connecting the input.

• Try to take, say 100 measurements in one cycle of the AC (20ms if 50 Hz, and 16.67ms if 60 Hz), and calculate the RMS then. (So sample at 1/5 ms - or 1/6 ms). If you don't, the output will never be stable. Ideally you should measure several/many cycles, so as to lessen the chance of noise interference.

• Take the mean of the measurements to determine what the center point voltage is.

Edit: They certainly did not burn out because of the 5V source! The power dissipated in both resistors is E^2/R, or 25/660000 = 0.037 milliW. They shouldn't even noticeably warm up. Even the smallest (even SMD) resistor should handle that without complaint. If they really burned out, then I would suspect a serious problem in your transformer. Personally, I would feel safer connecting the transformer to ground (instead of the divisor), and 'shifting' the voltage to the ADC range with an OPAMP. If something happens, the OPAMP will burn and not your processor.

On the precision front - I would not worry too much about getting the last bit out of the ADC. If you're doing 100 measurements, and repeat it over several cycles, you are artificially improving your precision too. I'd be more worried to have input voltage well in range - for safety. And put an R (say a couple of kOhms) in series for more protection. Something like this: