Hi,
1. I've never seen a case where the categories would show like that.
Try the "rebuild" button of the categories listing in the backend. That might help.
Otherwise, try edit your menu item, check its settings and save them. Or even, try to recreate the menu item with the same settings.
2. Regarding the cart, try the "check database" button of the HikaShop configuration. Maybe there is a problem with the database.
Otherwise, it could be some caching. Check that all the caching is turned off in the website, and on the hosting side (server/proxy, CDN).