Can I skip the lika and say hon är inte trött som han? Or maybe comparative construction demands this word to be present?
Yes, lika is necessary, otherwise it won't be a comparison of degree, as it is here.
Yes. In English it would be "She is not as tired as he is." The "is" is understood (and we would not say, "She is as tired as HIM is.")
I think PennLesley:s question wasn't about that (which is the suggested answer) but whether so should be accepted instead of as. And I'd say if you say so here in English, you should definitely say så in Swedish: Hon är inte så trött som han är.
In English class they used to teach (50 years ago) that it was 'as...as" after a positive but 'so...as' after a negative: : 'as tired as I am' but 'not so tired as I am'.
I think the 'so' is more of an American thing. I don't think I've ever heard it said that way in the UK
It is true that "inte lika" can mean "not as'" However, "olika" does not literally mean "not as" (simple negation), as far as I know. It means "various" or "dissimilar", and can be used as an adjective or an adverb.
So your suggested sentence, if it is not nonsense, seems to me to mean something like "She is differently tired than he is" or "She is tired in a different way than he is."
What do native speakers think?