I guess all of this is because you've figured that the thermostat is in the wrong place for your needs but you want to keep the features of the device (display, programming) without having to pull new wires from the heating system.
If i were going to do something like this, I'd go the @Idahowalker route and have multiple sensors. That's mostly because my bedroom can get very cold overnight but my thermostat is perfectly happy with what it's reading in the dining room. I think just turning the HVAC fan on would help me a lot.
As to relays, you will need something that will have a decent lifetime - from the price of the typical Arduino relay modules, I have a suspicion that those relays aren't intended for long term service, more like temporary hobby projects.
I'd take a look at your thermostat and see what relays it's using - you'll want something at least that good.
One precaution I've seen on other projects like this is to add an old school bimetallic strip thermostat in parallel with the Arduino controller. Set it to ten above freezing to protect your pipes if the Arduino solution fails.