Really good question, and others please weigh in on this as well!
I don’t think its the type of problems she faces (unless they’re always related to her being a woman) that make the treatment of a character misogynistic. I think it’s how she is allowed to react to them.
For instance, it would be misogynistic if 1) She has no control over how to solve the problem, 2) The issue is deemed too complicated for her to handle or understand, or 3) She doesn’t have agency in carrying out the solution.
I would add to this situations in which it seems that the female hero exists for bad things to happen to her. The only example that comes to mind right now (after I should have gone to bed) is Marion as she is portrayed in Robin Hood: Prince of Thieves (the one with Kevin Costner) and Robin Hood (the new one with Russell Crowe). SPOILER ALERTS for paragraphs below.
In the former, it seems like the only reasons Marion is even in the story is to 1) serve as a the love interest for Robin Hood and 2) to be almost raped by the Sheriff of Nottingham only for Robin to save her JUST IN TIME.
In the latter, however, Marion is a much more complex character. She is representative of the social upheavals in England due to King Richard’s obsession with the Crusades (in that she is left to run her father-in-law’s estates by her husband a week after they marry) and demonstrates the rage that many disenfranchised people felt against the crown because of this. She is also the person who introduces Robin to the injustices in England (since, you know, he has been at the Crusades this entire time, too); and where he tends to be sneaky and subversive in his approach to dealing with these issues, she is unafraid to bitch out the sheriff and the clergy who she considers to be unfairly robbing the people blind. Of course, in this version there is also an almost-rape scene (which I consider to be unnecessary and only in the movie to create suspense for the audience), but Marion very capably controls her fear and gets herself out of the situation rather than having to be rescued.
As a point of interest, this is a general point that I have heard frequently on Tumblr and elsewhere, not least in Anita Sarkeesian’s amazing videos on the treatment of women in video games. She includes a really on-point discussion of how rape and other violent acts against women are used to create suspense for the player in video games, which has a lot of relevance to movies and television as well. (In fact, that is a discussion that has probably happened elsewhere that I am not aware of, so apologies for my ignorance.)