I agree with Eli, I have never heard sick to the teeth, though in England we do say Sick to the back teeth a lot. I also agree although going nowhere argument is correct, it sounds very awkward and we would never say this. One good way to describe it might be, This is a fruitless argument. or This argument is fruitless. You may also call it pointless. We might also say that the argument is just going round in circles though this refers more to the idea of the same thing being said again and again. You also may like to say the argument is going down or is a one way street, there can be only one outcome.

I can only speak for myself, as a native speaker of English from the U.S., but I've never heard anyone say "sick to the teeth". I would say, "I'm sick of this stupid argument", or "I'm fed up with this stupid argument". I also think it's more common (and sounds much more natural) to say, "This argument is going nowhere" than to use "going-nowhere" as an adjective.

Yes, they're interchangeable idiomatic expressions 'a going-nowhere argument' is correct.