Electrical – How to fish thermostat cable using an existing cable

Ethernet cable should work fine since thermostats use low voltage and low current. For simplicity, it's probably easier to use each pair as if it were a single wire. That makes is easier to remember what is what at each end and clearer to anyone looking at it years later.

This sounds like an exercise in frustration. Even if you do manage to bond the wires together strongly enough, you will surely damage them by pulling them through staples or around corners. I have frequently seen damaged cables from being pulled through tight spaces. If you damage the outer sheath of a coax cable you will lose a lot of signal strength, and if you damage one of the wires in CAT cabling you will not get any signal at all (or it will drop down to 2 pairs instead of 4).

If you are really insistent on trying to pull the cables through without opening up the wall (which is probably what you'll have to do in the end), try this: get some tough wire pulling twine and tie that to the end of your old cabling. Then when you're pulling the old cable out you will pull the twine along to replace it. This will give you an opportunity to inspect the old wiring: if you damaged it pulling it out, you will surely damage the new ones going in. If you were able to get the old wiring out successfully, it will probably be easier to pull the new ones through with the twine since there will be less resistance. However I still think this is going to be very annoying and you'll never know the condition of the cables.