Learn English – What would be a a linguistic term for those nouns ending with -ing

If you're creating a neologism, I'd say go with the one that sounds the best.

The -hood suffix, to me, sounds more general and impersonal (or universal) while -ship sounds more personal. But that is entirely my own feeling.

Dictionary.com lists -hood as being "formerly used in the formation of nouns," so if you believe that means it isn't used anymore on new terms then that's an answer of a sort. But I think you're free to do what you wish. Hey, it's your word.

By the way, what's the word you want to create?

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