Why is Thermostat Y wire connected to C terminal on furnace?
Your AC compressor needs two wires - Y and C.  Y comes from the thermostat and C comes from the air handler or the common side of the 24vac transformer.

The picture you have with the mid-line wire splice is where the three conductor wire (green white, red) branches off and goes outside to the compressor.  Since Y is the standard for "compressor, someone decided to cut the Y wire in that place and use either end of it.  One side of the Y wire goes back to the air handler to pick up the C wire (T terminal), and the other side goes to the thermostat to pick up the switched Y wire.

So, the Y wire and it's connections seem fine.  What I would do is move the Green wire to the G terminal on the air handler and the thermostat since that is the standard color, and then use the Blue wire for C.  It can connect to the T terminal on the air handler and then the C terminal of the nest.

I think the Nest has a voltage check function, but a simple multimeter could be used to verify that you do indeed have 24VAC between the R and T terminals of the air handler.  Based on the diagrams, the T terminal is the C wire, and it was working before, so there shouldn't be a problem.