What is the correct use of "even" as an adverb, with the verb "to be"?

OED's entry for the usage OP is concerned with says...

Intimating that the sentence expresses an extreme case of a more general proposition implied (= French même). Prefixed (in later use often parenthetically postfixed) to the particular word, phrase, or clause, on which the extreme character of the statement or supposition depends.

It also goes on to say...

This use, now the prevailing one in Eng., is foreign to the other Germanic langs. It is rare in purely dialectal speech, and (though a natural development of 8) seems not to have arisen before the 16th c.

(OED's definition 8 refers to a largely obsolete usage where even = ‘namely’, ‘that is to say’, ‘just’, ‘nothing else but’, ‘to be sure’, ‘forsooth’).

Note that although they're very similar in meaning, there's a slight distinction between...

1: She even forgot my birthday
2: She forgot even my birthday

In #1, it's implied she performed other different actions besides forgetting my birthday (maybe she ate my last Rolo, for example). In this case, even modifies the entire phrase forgot my birthday.

But #2 more specifically implies that she forgot other things (not necessarily even relating to me).

There's nothing at all "odd" about OP's "Will you even be there?" (it's emphasising that not only might you not do something when you're "there" - there's some question as to whether you will be there at all).

OP's "Who even is she?" and "What even is that thing?" are highly unlikely constructions that would probably be considered unacceptable/substandard by most native speakers. The reason for this is that even never really modifies the verb to be at all - it modifies whatever comes next, as in...

"Can't you even be polite?" (If you can't be/act how I want, can you at least be polite?).
"I can't even be bothered to reply" (I can't do much at all, specifically not bother to reply).