I tried to do this and now I can't even access my website nor the includes folder on FTP
That doesn't make sense as FTP and the WordPress dashboard are 2 separate ways to get into your files. FTP should always work.
*Reads whole thread*
Permission denied in /homepages......./wp-admin/includes/class-wp-filesystem-direct.php on line 200
OK, permission problem there.
I have it installed through my hosting provider's app.
That may be the problem. Have you asked your host about that installer?
Could I just upgrade the files through FTP by downloading the latest version and uploading the files (except the wp-content)?
That ought to do it. Can you provide a link to your installation? If you can't or don't want to that's fine too.
Start by making a backup of what's there now including the database. This is just a safety precaution as I want to make sure something really bad doesn't happen.
Once that's done then do that re-upload on the WordPress 3.7.1 files skipping the
wp-content files. This time rename your existing theme directory to
wp-content/themes/your-theme-name-OLD and rename
This is just to make sure that your theme and or plugins are not interfering somehow. It's a temporary step (don't delete anything) and hopefully will let you into your WordPress dashboard.
If that does work, which theme are you using?