Is it possible to design a 2:1 multiplexer using only inverters, XOR gates, half adders and full adders? If so how?
X-Y problem why would you be thinking of a multiplexer given those components the examiners were giving you a clue. Expressing the problem in words.... If sign bit is set(negative) invert the value and add 1. If sign bit is not set don't invert the value and add 0. That should be easy with those components.

This is an xor on each value bit as an inverter which is switched by the sign bit. Then a half adder which adds the sign bit to the result of that.

If a multiplexer is deemed to be absolutely necessary you need an and function followed by an or function. The carry out of a half adder is an and, you can convert this to an or by inverting the inputs and outputs of the and. So you can make one mux bit with three half adders and three inverters.