Going from dual bowl sink to single bowl
I would not call a plumber. I would see if I could get a plug fitting for the slip joint nearest the Y and remove the left hand branch (no cutting or gluing). This frees up space under the sink. EDIT I couldn't find what I was thinking of, but there must be some fittings that would seal the 1 1/2" drain pipe.

You would connect what appears to be the dishwasher drain to a special tailpiece designed for DW connection above the trap on the right side.
DW tailpiece, e.g.,  http://www.homedepot.com/p/Everbilt-1-1-2-in-x-8-in-Plastic-Dishwasher-Wye-Tailpiece-C9816/205153966 or get a longer one if needed: Watts long DW tailpiece