Is it appropriate to say "Can you borrow me your ax?" instead of "Can you lend me your ax?" I hear the prior usage in the Upper Midwest quite often.
It's appropriate if the person to whom you ask the question 1) understands you; 2) would ask the same question of you if your roles were reversed; and 3) would not correct your usage by saying condescendingly "Do you mean 'lend me your axe?'"
Seriously, all people everywhere have both formal and informal ways of talking to each other. Who is to say what is or is not appropriate. Appropriateness is highly dependent on context. What is appropriate in one context may not be appropriate in another context.
Americans frequently confuse the terms lend, loan, borrow, and their cognates. It is correct to say, "I lent him some money," but not "I loaned him some money." The money that is lent is called a loan. "May I borrow your ax?" is proper, whereas "Can you borrow me your ax?" is, as you point out, region-specific and, while incorrect, is appropriate informally. A student, on the other hand, who asks an English professor, "Can you borrow me that textbook?" is simply looking for trouble!