I've never touched it beyond letting the default one work, but it's probably done using UIScreenEdgePanGestureRecognizer. If there's another gesture being grabbed by the active view (like scrolling) you might need to tell your UIGestureRecognizerDelegate to allow the gestures simultaneously.
EDIT - Oh I see, parent comment doesn't like the system standard of swiping from the edge of the screen. So that'd be a regular pan gesture recognizer, and probably trickier to do at the same time as scroll view.
I'd be hesitant to take that suggestion, since it means you'd never be able to use a "swipe right" gesture on the individual list items, like what you see in Mail for marking a message as read/unread.
Here's another feature suggestion, shamelessly lifted from the Apollo reddit client. Long press on the navigation bar to toggle between light and dark theme.
I'd also point out that your "sliders" icon for the settings tab is rather unconventional on iOS. Users will expect a gear icon to represent settings.
2. That doesn't seem complicated to add! Seeing the feedbacks here really gets me motivated to improve the app, so I'll try to add this in an upcoming version.
Thank you!