I recently used Apple Care support for an issue. They logged into both Mac and iPhone, but all they could do was guide me, they did not have control of the machines.
i imagine they can take control, but it is likely something you would have to grant. I believe they use the built in screen sharing functionality on Mac, which I assume is in iOS as well, but buried so users are blocked from using the feature.
So, most likely, they would guide you though steps to collect logs and you would send them for analysis.
Apple Care gets great reviews, I would tend to be pretty trusting of them getting into your system.
You can also try Genius Bar at a local Apple Store, they may be able to work with you in person to go through things.