Can I use the ATtiny84 ADC0 pin as both input and output?
There is a battle between the LM358 and the MCU pin so if the LM358 wins and it wants to output a fairly low voltage then the LED won't light up no matter how hard you try. I think a possible solution is to put a resistor (maybe 470 ohms) in series with the LM358 output. This will allow PA0 to win every time but the problem arises when the port is being used as an analogue input - will the LED and resistor affect the analogue voltage?

I'm not sure what voltage range you are using for your analogue input but if it's 1V maximum then you should be OK. Maybe OK even at 2V but above two volts the LM358 will be trying to drive the LED and losing voltage thru the 470 ohms I recommended you put in series originally.

If the Vmax for the ADC is less than 2V it should be OK and also add a 10nF cap down to ground on the input to reduce impedance increases from the 470 ohm resistor.