is it the 'de' here that makes it that she 'has doubts'? I wrote she doubts her husband but suspected something was off. But shouldn't there be a form of avoir here, and if not why?
French and English do not translate directly in this case.
"douter de" = "have doubts about" or "doubt about"
However, you can say "elle a des doutes sur son mari"
No, in English, possessive adjectives agree with the owner. Since the owner is "she", you have to use "her".
Yes, "she doubts his husband" would also be "elle doute de son mari", since in French, the possessive adjectives agree with the thing owned.
have you guys just written opposites? how is it then with she and "his husband"?
It depends how the exercise was presented:
Dictation in French: "son" is most probably "her" husband
Marginally, it could also be "his husband" (Isabelle doute du mari de Pierre). But in France, please wait a bit since mariage between people of the same sex has been legal only for a couple of months ago!
- extra meaning: even with "her husband", it could be another woman's husband (Isabelle doute du mari de Jeanne). Context would tell.
Written French for translation into English: see above
English text for translation into French: only "her" (+ extra meaning)