I just watched an American TV show and a guy say you'd have to lay on the cold floor and stuck your fingers.

Actually, I thought 'would have to' means potential future but in this sentence, it is not used that way.

if this sentence is simply changed into 'used to lay on the cold floor', they still have the same meaning?

and when do you usually use would have to expression?

See a translation

It is potential future. But here, it's saying 'if you want to achieve your goal, you need to do this.' The goal is in the future.

Here's some examples:
"You'd have to steal if you wanted money from them."
"You'd have to lie if you want to get inside."
"You'd have to study if you want to pass the test."
"You'd have to eat this if you want to be healthy."
"You'd have to carry me if I go with you."
"For this plan to work, you'd have to pretend to be dead."

I hope this helps! ^-^
What TV show were you watching? Because the context may be a little different. (And I can't answer the second question without knowing T-T )