Hello @HoganVanSickle ,
At this time there isn't exactly a way for the app to prompt you to open Google Chrome when accessing external tools or links. When you click "Open in Safari" or "Launch External TooL" from the app, it will automatically populate in Safari. However, if Google Chrome is the default browser for your iPhone, when click either of those buttons, you should see the "Open in Browser" icon which looks like a compass. This involves one extra step but that's how you would be able to access your course content in Chrome from your phone.
As far as syncing your Canvas Calendar to your Google Calendar, that would need to be done from a computer/browser.