Help installing new thermostat
The red wire that is currently on the C terminal, runs out to the coil on the contactor in your condensing unit (A/C). You'll notice the white wire from the same cable is on the Y terminal, which connects to the other coil terminal on the contactor.

Technically you shouldn't put two wires under a single screw terminal, but it's done quite a bit with HVAC control wires.  Simply put the blue wire under the C terminal, and connect the other end to the C terminal on the thermostat. Then you should be in business.

Based on the information you've provided, the wiring seems to look something like this...



The blue dashed line should represent the unused blue wire in the cable.

The installation instructions for the CP-22 condensate pump, shows the unit being wired off the Y wire between the thermostat and the furnace. 


  


Presumably, this is so the pump knows when the A/C is on. In your situation, it's wired off the R wire between the furnace and the thermostat. I'd speculate that the installer wanted the pump to always be active, as long as the furnace had power.

I can't say for sure that this is what's causing the problem with the C wire not working, but it's definitely something out of the ordinary.  Without a multimeter, there's not much troubleshooting that can be done.  If you had an HVAC technician on site, they would likely be able to install the thermostat. Unfortunately, it's difficult to do this type of work over the internet.