Electrical – use the unused blue wire as a C wire

Based on the second image, I'd say TR2 should be connected to the C terminal. However, for this to work, you'll likely have to swap TH1 and TH2. Since TR1 is associated with TH1, TH1 should be the R wire while TH2 should be the W wire.

The complete circuit should be:
TR1 -> TH1 -> R on thermostat -> W on thermostat -> TH2 -> TR2.

Then you can connect a wire from TR2 to the C terminal on the thermostat.

NOTE: If the transformer is not sized to support the additional load of the thermostat, you could end up blowing a fuse or frying the transformer.

There should be no problem connecting the C wire from your thermostat to the C terminal in the furnace. Take a look at this diagram, which is a rough approximation of your system.

Notice the cable going to the condensing unit has a red wire connected to the Y terminal in the furnace, and a white wire attached to the C terminal. When the thermostat calls for COOL, it does so by energizing the Y terminal (connecting R to Y). This provides power to the coil of the contactor, which allows it to close the contacts and turn the condensing unit on. The other side of the contactor coil is connected to the C terminal, as to provide a complete circuit.

You'll notice in the first image that two relays in the furnace, the thermostat, and the contactor coil all connect to the same leg of the transformer. This is because the thermostat basically works like a switch, connecting the R wire to the C wire through control circuitry. Anything that needs 24VAC power, will be connected to both legs of the transformer.