No, that definitely puts her there in constantinople. You can in fact do things like spread disease this way. There's a few other instances in which character location is important, I believe you need at least one character in the same province as the target of an assassination plot to get an opportunity, regardless of your plot power. According to the wiki you can also avoid disease by raising an army and leading it somewhere where the plague isn't ravaging. Not sure I fully believe that though, I think as a ruler you might always be considered as occupying your capital unless you're on one of the specific event-based trips that give you a regent.
Nah, even as ruler, you travel with your army. It's why you can die on the battlefield. And why the scurvy exploit works. Do you have a character in your kingdom that you wanna kill? A vassal, heir, or even yourself? Raise levy, add him as commander, load the levy to a ship, set sail to sea, and just park it there. After a couple months, the character will develop scurvy and likely die soon.