Hi
Regarding your first question:
First of all, there is a feedback loop even with no load as you are connecting the output back to the input directly and the feedback factor, in this case, is one. I think that you need to check the range of the input voltage (Common-mode input range) of the EA and check if still provides a good gain (or it can be better if you lower the input voltage). Given that you are designing an LDO, then the output voltage is in the range of 200mV less than the VDD.
With that said, I think you need to do stability analysis because one main challenge in designing the LDO (with PMOS pass transistor) is that it conventionally has two poles one at the gate of the pass transistor and the other one is at the output of the LDO (you can do transient analysis with a load current step and with no compensation you will be surprised with the output voltage !)
Back to the question of the 3% problem, I think that the current is so much low that the EA because of the FB (if designed correctly as negative feedback) is trying to push the gate voltage of the PMOS as high as possible. As a result, the output stage will have transistors that go out of saturation and that kills your gain!. Thus, the Loop gain at this condition will be very low and the EA has a large error between the reference voltage and the output voltage in doing the comparison.
The second question: You don't need to waste 1% of the max load current. However, you need to maintain the transistors in EA in saturation at the two extreme conditions the maximum load and minimum load.
Question 3: I am not able to help without knowing the topology of EA you are using.
You can use an extra stage at the output of the opamp to drive the PMOS (More effort should be done in this case with stability )
Best regards