Even if an American was not born in Japan, they should have somewhat of an accent when speaking Japanese. I've been told, it is helpful (because using the accent of the language you're speaking in just makes everything slightly easier to understand). And some Japanese words just *have* to be said a certain way anyways, so they might have an accent. I'm really sorry my answer was so long, I just wanted to explain well. Hopefully that makes sense!

yes. unless they are at such level that they do not have accent anymore.