you can edit the slug in Settings > Pages > Permalink Slugs
but you’d probably not be able to have both events and categories use the same ‘events’ slug, as you’ll probably run into clashes (never testd this though, might work, try it!)
I use “events/current” for the “events” slug and I use this plugin: http://pastebin.com/NGqWVJt8 with a little modification to change that slug for single events to “events/YYYY/mm/dd” (I replaced “$event_slug” simply to “events” in the code.).
This way I can set the “categories” slug to “events” in the Permalink Slugs settings and it works as expected for single category pages. But this way the category list page “events/categories” doesn’t load. Would it be possible to set the slug to “events/categories” and use a plugin like the one aforementioned but for categories?
interesting, never saw that snippet before
you could make this work but I wouldn’t be able to help with this, needs tinkering and testing