What are noise and accuracy improvements that can be made to the classic "error amplifier + transistor" linear regulator?
All linear regulators are similar to this- there is noise due to the reference and due to the amplifier. The popular band-gap reference tends to be noisy. A buried zener reference is typically less noisy but requires a relatively high voltage. All references drift with temperature and time, and there will be some effect from the power supply voltage. 

There is some error due to the offset voltage and finite gain of the amplifier, as well as the reference. Adding an integrator could make the error disappear, at the expense of some stability. The amplifier will also have some effect due to changes in supply voltage (line regulation).