Learn English – the gram­mat­i­cal term for “‑ed” words like these

As a personal opinion, I'll suggest that there is no tidy single word term for gender specific words. In fact "gender-specific" or non-gender-specific or similar are probably as good as you can get.

No noun is immune to gender differentiation, or removal of gender differentiation. If somebody takes a term that is usually asexual and produces two new gender based variants that are recognisable, it will probably not survive, but there is no reason why an especially apposite creation may not become part of the language. Some words may have a general and eg female version but no distinctly male equivalent. eg the (contrived for the purposes of this illustration) term "pilotess" would be immediately understandable. I cannot think of a distinct male equivalent. [For female-er-ising, addition of "-ess" works in many cases ! :-) ].

Such choices either way, as have occurred in 'recent times', are liable to have been driven by the desire on the one hand to use gender inclusive language, and on the other hand to use terminology which makes a point about discrimination or differentiation when it is used.

The term dancer is indeed gender inclusive, but "ballerina" exists as a term which overwhelmingly suggests a female protagonist. The term "male ballerina" gives 14,000 Google hits - but most seem to be asking what the correct term is (Some suggest "Cavalier"). Some sites such as this one are so bold as to use the term directly , but still manage to revert to the occasional "male ballet dancers" indicating that the usage is unusual.
The suffix " ...ina" tends to suggest either 'small' or 'female' but this is not necessarily so in all cases.

For added fun, consider the gender inclusive / male only / female only versions of: Waiter, Host, Bellboy, Pointsman (cars), Point-man (guns), Aviator, Dominatrix, Seamstress, Druid, Governor, Best man, Minx, Cougar, Priest, Nun.
Many need extra discussion to explain variants. eg "Dominator" may be the proper male version of Dominatrix but loses a certain something. Priest may become priestess, but not always. Seamstress seems to have no equal. etc

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Is it ever possible for a sentence to have a word in it
that is simultaneously more than one single part of speech
in that sentence, under the same parse and meaning?

So, if a grammatical English sentence contains a word A, can A be more than one POS?

Parts of speech are grammatical terms and have varying meanings for different grammarians.
Let's rule out quantum superposition of POS, so no Schrödinger's Gerund that's noun and verb.

There certainly are sentences where it's impossible to tell which of several possible categories a word falls into, like the first sentence below, where exhausted can be either a predicate adjective, as in the second sentence, or part of a passive construction, like the third one.

• I was exhausted.
• I was exhausted and the bed was soft; we suited each other well.
• I was exhausted by the irritable conversation and left early.

But that's not "in the same sentence". In the first sentence, there's just no way to know what the speaker intends about POS; it could be either one. And there's no way to know if one speaker might feel it was an adjective, but another speaker might think it was a participle. Or the same speaker might do both, to the same sentence, on different occasions.

So, the key word in the question is Simultaneously. And the answer to the question is No.

If anything in a parse changes from one POS to another, that makes it a different parse.
Thus, if A has two different POSs, they will occur in two different parses of the sentence.
And therefore the sentence can't simultaneously have two POSs in the same parse.

It is of course very easy to find sentences that have two interpretations; this is one way to make jokes, and certainly such ambiguity is common. But each interpretation represents a different parse. That's one of the purposes of parsing, in fact -- to distinguish ambiguous sentences and make their differences distinct. But that doesn't mean they're simultaneous, in any sense.