It's fine. The language is formal and rather literary, but perfectly correct. We often come across this type of sentence structure in quotations from texts written in centuries gone by. We would not use this style of language in modern everyday English unless we are quoting a proverb of some kind. NB The word 'so' represents the word 'happy', and it has to follow the word it refers to: "...think himself so" means "...think/consider himself happy". You cannot reverse the order of these two words.

It's correct. "So" refers back to the word "happy." "To think oneself" something may be a little unusual in today's spoken language, but it's still perfectly correct and usable. We could also say "to think of oneself," or "to consider oneself."

I'd put "...who doesn't think so himself", sounds more natural to me.