The standard gender-neutral pronoun in Finnish is "hän" = "he/she", plural "he" = "they".
This really doesn't have that much effect, as far as I can tell. A language with non-sexed pronouns doesn't seem to change culture much. There are still lots of male/female words, including the female suffix "-tar", roughly English "-ess", and compound words like "palomies" (fireman), with the same debates about changing to gender-neutral forms. The rest of the language apart from the pronouns is as sexed as any other. Finns do know what a woman is!
The Finnish word for sex is "sukupuoli" - literally "family half", which I like. They don't have a word for gender exactly - a translation would be "sosiaalinen sukupuoli" - "social sex".
What I find amusing though is that Finns can't actually be bothered with the person/object distinction and actually use "se/ne", which are formally "it/they" for objects, instead of "hän/he" in most spoken dialects. So people are usually called "it".
(You can observe Finns speaking English fairly often will stumble over getting he/she right, but usually spot the error and correct themselves).
I've not actually heard about the neopronouns - sounds hilarious.
I had heard trans people complaining about "misgendering". If people don't hear other people calling you "she", how are they supposed to know you're a woman? Lack of sexed pronouns makes it harder to tell someone is supposed to be trans. 