What does "cold balls" exactly mean in American english?

I think you're putting a bit too much emphasis on the word "cold". Although it could have some connotation of emotional coldness here, I wouldn't say the overall sense of the expression emphasizes that the person has a cold heart.

I don't think you'd ever hear "cold balls", as in your title. The word "steel" is an essential element of the expression in the dialogue that you quote.

I would say this is a variant of an expression where a person is said to have large or tough balls (=testicles) as a way of describing courage (as you put it) or boldness or fearlessness. Basically, "steel balls" = metal balls = unusually tough balls = an unusually large amount of courage.

"Cold stainless" is an extra bit of description added before the word "steel". There could be various reasons for using those particular words: it might be to add some reference to the "coldness" of the behavior being described here, but it could be to make the expression sound better or more striking. These kinds of expressions often get creatively modified from a basic format: e.g. I think "brass balls" is a more common form of the expression.