What you describe isn’t a good sign, and, for me, would definitely call for a re-install of WordPress.
1. Backup your site to somewhere safe and easy-to-restore-from (like VaultPress).
2. Others may counsel against this, but t’were it me, I’d give that reistall button a go and see if it grabs 4.1.
3. If not, or if #2 messes up, or if you want to be safe and skip #2, what you need to do is reinstall manually:
http://codex.wordpress.org/Updating_WordPress#Manual_Update
It’s not hard, but it’s a bit precise, and you need to be careful about keeping your /uploads folder (and really, everything in /wp-content). But if you’re not an FTP kinda gal, then maybe this isn’t for you (or hire someone who’s comfortable FTP’ing).
Another possibility is your webhost may offer a way to upgrade WP from their admin panel: many have this feature.
Then, once you do get your WP to 4.1, I just know, from now on, you’ll upgrade as each new version is released so this unpleasantness never happens to you again.
There could be something strange going on with your site, but it might also be that you’re on managed hosting.
If you are certain managed hosting environments, you can ONLY upgrade your WP version through the hosting dashboard. Note also that most managed hosting includes a bit of caching, so after updating your site, you will need to flush the cache so that you can actually SEE the update. If you’re on managed hosting, contact the hosting company for help (after all, isn’t that most of the point of managed hosting?).
If you are updating more than 2 versions, it’s recommended to do a series of smaller, manual update – i.e. from 3.5 to 3.7 to 3.9 to 4.1.
Detailed info here:
http://codex.wordpress.org/Upgrading_WordPress_Extended#Upgrading_Across_Multiple_Versions
It’s also quite possible that your theme will have problems with an updated version of WP. So you may need to deal with fixing it or finding another theme.
But the biggest reason to get updated is that using an outdated version of WP is a MAJOR security risk – especially one that old. If your site gets hacked, you’ll have a much bigger problem than SEO :)!