Is Research Fellow a gender-neutral term?

Yes, in your reference, the third definition is the one being used, rather than the first:

a member of a group of people who have shared interests, activities, etc.

Of course, we should look to how the word is actually used rather than solely to its dictionary definition, but in this case I think that actually makes the argument even stronger, since academics use the neutral form quite a lot and nobody else (at least in my part of the world) seems to care for the word.

Yes, "research fellow" is a gender neutral term, just as Simone de Beauvoir can be called a "fellow traveler".

The word "fellow" derives from the Old English feolaga which means roughly "one who shares something" and is etymologically not gendered; you are being misled by the more recent colloquial usage (less than 600 years old) to mean male person. But the meaning here, which is specific to the academic context, developed separately (via the notion that the fellows of a college share in its revenues). You can find more details at the wonderful reference, the Online Etymology Dictionary.

Indeed, as you suspect, etymologically it is not gender specific. If you think about the usages, fellow typically just carries a connotation of "going along with" or "having in common with." See this page for more on the etymology. The wikitionary entry also has usage notes stating fellow is not typically used in the sense of "a man" in North America.