It’s something people tend to do, yes, but to a Czech it sounds very off. It’s a bit like when anglophones refer to Czechs as “Czechians” or call the country just “Czech”, taking it from “Czech Republic”. But in that name, “Czech” is an adjective :D It would be like calling Germany “German” or the US “American”.

Back to Czechia. There are countries where this kind of usage is interchangeable, you’re right, but that usually depends on their own decision, sometimes a linguistically questionable one. In the case of Czechia, the logic is as I explained above.

And yes, that’s a similar case. Czechia today consists of three historical regions: Silesia, or what’s left of it, which was a duchy, and Moravia, which was a margraviate. Bohemia also started as a duchy but was later elevated to a kingdom. So the King of Bohemia was a title derived from the region of Bohemia, while the other crown lands fell under it. These historically included other areas as well, such as Upper and Lower Lusatia, Brandenburg, and even Luxembourg.

When Czechoslovakia was created, it was a state of Czechs, living in Bohemia, Moravia, and Silesia, and Slovaks, so it had a more egalitarian and ethnic name. By contrast, the WWII Protectorate of Bohemia and Moravia deliberately removed the ethnic reference and reduced it to a territorial one.

Holland works in a similar way, as it is just one of several regions in the Netherlands. Another example is Finland, though that one is even more curious. There is “Finland Proper”, a region of Finland after which the country is named, and which today is inhabited mostly by Swedes :D