Yes that's a controversy at the moment -- some people refer to "Ukraine" (the country) and others refer to "the Ukraine".

I think the word "the" implies that the place that's being named is a region, and is not a single place or a single political entity -- so "Ukraine" is correct unless you deny that it's a country.

So "the Himalayas" -- or "the Caribbean", "the Great Lakes", "the Prairie Provinces" -- but conversely "Mount Everest" or "Jamaica" or "Lake Superior".

Or in your first example, "the city", there "city" is a common noun and not a proper name at all.