Is it appropriate for an American Ambassador to be a spy?

Maybe not appropriate but it happens. The Soviets were famous for it.

There is really nothing ethical or unethical where intelligence gathering is concerned. As an ambassador, or member of his staff, he is required to represent the best interest of his country of which he is working for. but, there is always a trade off. If he spys and gets caught, it will hurt the credibility of his countries integrity and what he gained from spying may not be more valuable.

It is not appropriate. The US was famous for it too.

It got the US diplomatic corp into a lot of trouble in Iran in 1979.

I don't see any reason why it would be a problem, although Ambassadors are usually well-known people who have had political careers throughout their lives.

Intelligence comes from trained professionals.

Diplomats are not trained professionals, but they can certainly transfer information that can help protect Americans.

Shouldn't they always be looking out for things that can help America and stop things that could hurt America.

Yes, its their job.

No, there is not any "ethics".

Imperialism ignores "ethics".