Are idioms and paradoxes the same thing?

No. While an idiom often means something other than the literal meaning, a paradox would be a situation that is really opposite of what is expected. For instance, gas and air valves often leak because of debris in the valve. If there were debris in the valve, you'd expect it to be stopped up. That would be a paradox. However, saying that the drinks are "on the house" would be an idiom. There are no drinks on top of any house. It just means they are free.