These are "assumptions" that almost zero real people actually assume.
No, that's pretty much the default assumption of the statist. It sounds like you just don't like the term.
You can't. The word is used solely by anarchists to mean "everyone who's not an anarchist", except that the left-anarchists decided they had more common cause with non-anarchist socialists and dropped it. Thus, the only people using the term "statist" of their own initiative are minarcho-capitalists, who use it to mean, "Everyone who's not us."
There is no such party as the Statist Party. There is, however, such a thing as Every Party But the Libertarian Party, which is what "statist" actually means.
The word is thus empty of specific semantic meaning, and is simply a smear-word used by minarcho-capitalists.
And you're wrong about only anarchists using that term. But I find your minarcho-capitalists usage pretty amusing.
Hence why leftists aren't called "statists" by the way: because the Left includes anti-state, anti-property leftists (who see no contradiction there: they hold that it's the state which creates property in the first place and property arrangements which create the superstructure in which the state acts!).