How to move utility sink and extend trap / drain pipe under it?
You would do well to incorporate an elbow or elbows, if possible into the part of the drain line that is behind the white PVC pipe, the metal pipe. The trouble with that would be is that old of pipe is easily damaged (broken) at the joints where the threaded sections are.

The reason I suggest this as the place to tie in the new extension is that by the time, I feel, if you simply replace the drain it will be so far away from the wall it will be a number of inches away from the wall it is against now. To remedy that, it would take so may elbows to get it back to the wall, I am afraid the drain would not drain readily or perhaps clog easily. So back at the reason why I mentioned taking it back and tying it into the farther in from the joint where it is tied in now. The place I refer to is at the wye, there is a small straight pipe there, perhaps there is a smooth part (no threads) to it to use a neoprene coupler there.