Yes, being of European descent is the definition of white.
For a start, no (Are Arabs non-White? What about, ehm, Caucasians?)
Second of all, the definition of White in the 1920s would have necessarily have included Southern Europeans.
Basic rule: All White are Caucasians, but not all Caucasians are White.
I probably should have said that was the standard European distinction, which is somewhat modified in the Americas (but I think less so in practice than officially). Latin America has the broadest definition of white.
1) In Europe Arabs are non-White no matter their skin colour. This is different in Latin America and maybe also partially in the US (I am a bit sceptical as to whether that is really the case - it seems often to be a mere formal recognition - "they are white, but not white-white". Christian Arabs has generally been considered whiter than Muslims. At least some Australians consider Lebanese as non-White and I get the impression Austrian racial terminology is (still) pretty much the same as the standard European, but not sure.
2) Caucasus is a part of Europe, but whether they are socially considered Whites varies a great deal from place to place. I assume Armenians and Georgians would be considered Whites in most contexts by virtue of also being Christians. There has been a clear tendency to consider people from majority Christian countries as White, while people from Muslim populations with roughly equal skin colour are non-White (Greeks are White, Turks are non-White).
3) Not sure about the meaning of your second sentence, but assuming you mean "wouldn't necessarily have included" that was never the case in Scandinavian racial terminology and obviously not in Southern Europe itself. It was more common to omit Iberians and other Mediterraneans in English, but far from universal (but you obviously know more about that). Regarding the Spanish Civil War classicism and hatred of the godless reds + revenge for leftist atrocities against clergy were more than enough reason to treat ordinary people with cruelty. I think the "non-White factor" was fairly marginal and exaggerated. Would be interested in sources that say otherwise, of course.
Anyway, eliminating Southern Europeans as non-Whites must be a thing of the past in Europe. I think the EU and immigration from other continents has contributed to considering all "natives" as Whites. In Latin America this was never a thing. Italians and Greeks are whites in Australia. Then we have the US where the Hispanic label confuses things regarding Iberians (which was our departing point). Still, Italians and Greeks are obviously whites in America.
tl;dr: race is a social construct and it varies a bit, but a person of European ancestry is the basic defintion of White in all Western countries.