The misspellings seem to be triggered by various uncommon punctuation. For example,
Plain Text is actually more of a problem here than a help, since this isn't plain text, but a markup language. See, plain text does not create scopes for different syntax, so tokenizing is done on high-level english punctuation. Normally
^, etc are not part of prose.
If you use the correct syntax definition with the markup, and the syntax definition properly defines tokens, the spell checker should receive usable words, and should give accurate results.
If you provide some examples of misspellings in an AsciiDoc document, I should be able suggest tweaks to the syntax that would help.