My attempt:
whether the phrase "этонесутьважно" is grammatically correct,
No. But yes, it's a relatively widespread phrase. I believe I've heard people saying it.
if the phrase is incorrect, why people use it that often, strongly preferring it to "этонеестьважно,"
They don't (usually) realise that it's grammatically incorrect and/or don't care. They just like using this phrase and use it (knowing that they will be understood). See more on this below*.
what people actually want to say with the phrase, if different from (4)?
They mean "этоневажно", "этонеоченьважно", "невэтомсуть"...
*- Basically, this is "невэтомсуть"+"этоневажно" (upd1 probably also influenced by "нестольважно", which sounds similar) glued together in the sayer's mind. I struggle to give a proper term for such a language phenomenon off the top of my head but the gut feeling tells me that such a term exists, as well as other examples of the phenomenon. Intuitively, I would call it a variant of tautology.
"Этонеестьважно" is not used because "есть" is strongly felt as being redundant here. But some individuals could say that. More often you might hear the phrase "этонеестьхорошо". These are the examples of … well, some people just want their speech to be "with a twist", they just like "breaking the rules" slightly. Thinking, probably, that this makes them more interesting or sound a bit more clever. When such phrases are overused I personally find this rather annoying and silly.