That sounds as if it could be a browser cache issue. It commonly happens with updated CSS files, the browser uses the old version from its cache, not knowing the file has been altered.
Try clearing your cache, or add a fake variable to the file URL so it loads the new version:-
<link rel="stylesheet" href="styles/my_css_file.css?version=2018">
That sounds like a problem with Google's index not updating yet. It should happen naturally over time but you can do things to help. Submit your new pages to Google using "Fetch as Google" in Search Console. Set up redirects from your old pages to equivalent new pages.
That does actually sound really odd, I've never seen that, but it may be due the the caching issues I mentioned before.
Any reference to a file with the exact same name and URL as before, but has changed can load an older cached version.