Sensing Voltage and Current of an Adjustable Voltage Source
It is possible to figure out the voltage and resistance of the load by
  knowing the current and voltage across the shunt?


Yes, if you know the voltage across the shunt, it's a simple V/R=I equation to know the current. Since the input bias current to the op amp is infinitesimal compared to the current flowing through the shunt, the current flowing to the op amp can be neglected. 

As far as the voltage goes, you can consider the voltage out to be the same as voltage in, with a few exceptions. Vout can't get as high as the rail, the voltage drop across the darlington pair will limit the voltage to something like 14.5V or 14.7V. The load will also limit the voltage range of vout, the lower resistance, the lower the voltage the circuit will be able to maintain. 

For example, the darling ton pair saturates at 1A, and you put a 5Ω resistor  for a load, the circuit cannot supply more control energy beyond 1A, so you would be limited to range of 0 to 5V. With a 10Ω load, a range of 0 to 10V. 

It will be a good idea to select your components and run a spice AC analysis on the circuit, the capacitance of the transistor pair can change and create oscillations at high frequencies. A good place to start is an RC filter between the negative terminal and Vout. A series resistor on the output of the amp to limit the current is also a good idea. 

These circuits also can have problems with inductive loads, if your load is inductive, the feedback loop will need to be compensated for it.