I am concerned about the weight loss and vomiting. The hair loss - it is difficult to say without seeing the patient. The hair coat on pets is highly sensitive to internal stresses. It may also be unrelated.
My recommendation is to pursue further workup on this guy. Whether the hair loss is related to the tumor/liver/wt loss or not; further workup is definitely in this cat's best interest and if it IS related - then you will find out as he is worked up.
The good and bad new with livers is that they can compensate for a lot of damage which means that the pets can be looking better on the outside than what is really going on, on the inside! If the tumor has spread within the abdomen, there may be more options to maximize quality and quantity of life before more clinical signs appear. It is also possible that during you find out that there is something non-cancer related. I am concerned however and recommend that you continue on with your veterinarian's advice as soon as you can.
Good luck and keep us posted.
Christopher A. Lee, D.V.M., C.V.L.S.
Promoting surgical options and pet comfort through the use of lasers.