1) Current regulation gain is too high and saturates to ground. The loop gain and phase margin is important for step load stability too.
2) If can create -2.0V say using some other source then you connect the R divider Vadj to drive the output to 0V.
RED LEDs are good for 2.0V near 10~15mA and you might consider a charge pump inverter or some other convenient source.
The problem with this method is LDO"s are series pass and high impedance at 0V out , so a shunt regulator would improve this on the output. Essentially a Half bridge . But very inefficient. Thus a SMPS regulator is more efficient. There is no simple solution for your wide ranging requirement.