Well, the are two sides to the answer. It's either the exam went well or it didn't. If it didn't, I am not sure your tutee will be comfortable to tell you about it. Another thing is that your tutee may also expect you to ask how it went, to show you care. So it depends. Regardless though, it's not a bad thing to ask a student how the exam went.
As a tutor in undergrad, we were taught to ask "how did the exam go?" and not "what did you make on the exam?" The idea being that the tutee gets to project their feelings onto their score (some are happy with a 75) and gives them a way out ("ehh ok I guess").
It's implicit in these instructions that asking "how did the exam go?" is appropriate for the ...
I expected her to inform me how the exam went
I'm not sure why you expected this. While it's certainly not unusual to follow up with a tutor (or thank them), it's not a requirement.
Is it appropriate to ask her how the exam went?
I see no reason why not.
Many university professors have side gigs doing consulting work for industry, writing, starting companies and doing many other things. At all universities I’m familiar with this is permitted and to some extent even encouraged. Usually there will be a policy in place specifying how much outside work is permitted. For example, here is a link to the relevant ...
I'll just have to guess that most places will have regulations concerning, or even forbidding, this. But it is a big and variable world.
But, at a minimum, a person contemplating doing this needs to check with the university administration, who may permit it or not. Or they might set some boundaries. I doubt that many would permit tutoring students at the ...