Will having reversed hot and cold damage my faucet?
The biggest risk is someone being burned.  In North America, cold should always be on the right side and hot on the left.  If you are outside of this region you should check with local standards.

It sounds like you either bought a faucet from a different region, or maybe your original faucet was installed backwards.

With regards to the actual plumbing, it shouldn't make a difference which side you are hooked up to.