lo.logic - elementary equivalence of infinitary symmetric groups

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 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.]