Is “misogyny” only applicable to men? What is the antonym of misogyny?

I'm afraid your etymology is incorrect: it comes from misein "to hate" and gunê "woman".

In your quotation, it is not Liz who's misogynist, but the story in which she figures, "all this": the script writer portrays her as a stupid woman, and the article is telling us that, by so doing, he is disparaging women in general, in an anti-feminist manner. It would have been less confusing if the writer of the article had removed the word "and" while adding a full stop after "her job": then it would have been clear that "misogynist" refers to all the preceding.

The antonym would probably be someone who likes women. That would be a philogynist (from phileô, "to like"), but I don't think that word exists. There is also a philanderer, a man who likes to court women a little too much. This word is a bit strange, since ander- comes from anêr, "man".

A man-hater would be a misandrist perhaps, or a misanderer—but I don't think those words exist either, or at least they are rarely used: it is just not an historically significant concept.

You've misconstrued the etymology of misogyny - it's Gr. miso (I hate) + gyne (woman). It's only applicable to "woman-haters".

The equivalent for "man-hating" is misandry. Hence misandrist n., one who hates men, a man-hater (esp. in feminist usage).

I also think you've been misled by the conventional assumption that all misogynists must be men. It seems to me your correspondent is suggesting that Liz Lemon's inability to trust her own judgement, etc., is being portrayed as a manifestation of her (Lemon's) misogyny. She doesn't trust herself because she's a woman. But the logic is confused because Liz Lemon is a fictional creation. It probably makes more sense, as Cerberus says, to lay the misogyny on the script-writer rather than the character.