For question 1, it must be true for huge numbers of cardinal pairs, for the simple reason that there are only continuum many first order theories in a countable language, but there are more than continuum many uncountable cardinals. Thus, in fact there is a proper class of cardinals serving as positive examples of your phenomenon.

The same idea answers part of question 2. We have essentially a map from pairs of cardinals to the corresponding theory, and since there are again only continuum many theories, there must be many a proper class of pairs of cardinals getting the same theory.

I am expecting that they are all elementary equivalent.

[Edit: my expectation is apparently refuted by Mckenzie and Shelah.]

1

For question 1, it must be true for huge numbers of cardinal pairs, for the simple reason that there are only continuum many first order theories in a countable language, but there are more than continuum many uncountable cardinals. Thus, in fact there is a proper class of cardinals serving as positive examples of your phenomenon.

The same idea answers part of question 2. We have essentially a map from pairs of cardinals to the corresponding theory, and since there are again only continuum many theories, there must be many pairs of cardinals getting the same theory.

I am expecting that they are all elementary equivalent.