Safari in a strange mode where pages randomly zoom in and pages slide to the right
Sounds like you have system zoom enabled. You can disable it from System Settings > Accessibility > Zoom.

When you have system zoom enabled, you can toogle it with cmd-opt-8, or use your control gesture which is by default the ctrl button in addition with the scroll gesture (i.e. sliding your finger across the mouse as you mention, or using the scroll wheel on a mouse).

Regarding Safari it seems like you're not aware that you have gestures enabled. If you go to System Settings > Trackpad or System Settings > Mouse according to what you use, you can see the gestures that are enabled. Check for "More Gestures" and you'll see that for example "Swipe between pages" will look like pages just "slide off Safari", where you're really just navigating back/forth in the history using gestures.