Question on Furnace/EIM Wiring and Splice
Your system is not really hooked up wrong, just odd. It looks like the installer read the directions from Honeywell and it looked like he needed to take power directly from the transformer. Honeywell doesn’t know what equipment you will be hooking up to so their directions are generic. Those two wires from the transformer do not need to be there. The R from the furnace should go to the R on the receiver and then jumper to Rc and Rh. The C from the furnace should be hooked to the C on the receiver. The extra wires are not hurting anything since they are coming from the same 24 volt transformer. 
If you were to install a new thermostat you would need to remove the old thermostat and receiver since the receiver will only work with either the Focus Pro and Vision Pro wireless thermostats. You would then either run a new  5 wire cable from the furnace to the new thermostat or run a cable from the new thermostat to where the receiver used to be and splice into the wires there.
I would use the blue wire to hook to the C on the furnace and the C on the receiver and loose the extra cable if you were to change thermostats or just leave it since it isn’t hurting anything.