Yes. "Other country" will work fine. If your state has specific country codes and you file by mail you can white out and enter those manually.
That's not necessary. I suspect the specific codes are for state record keeping. States almost never have tax treaties with other countries, so your nonresident spouse's specific country of residence will not impact the state tax return.
You can find a summary of U.S. tax treaties at United States Income Tax Treaties - A to Z.
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