It was a database problem, one of the mods added 'spam' field with no default value to _topics table. It should work, but didn't. I believe errors were caused by a mysql bug, it doesn't affect core code, but causes problems for some mods. I set default value to 'null', stopped the errors and shouldn't affect mod.
Package manager creates a backup whenever a mod is installed, they're in /Packages/backups/. If you install/uninstall/reinstall mods, things get confused and backups may not undo damage.
Your cpanel has a backup utility, but it doesn't allow automatic backups. Most hosts do daily server backups, but many only save backups for a few days, then they're gone. Some will charge you to restore files or databases from server backups. You should confirm with your host how long backups are saved. My host uses R1Soft, which backs up all files and databases daily and saves them for 30 days on remote server. I also have cron jobs running backup daily and save 30 days in my account. Your cpanel supports cron jobs, but you don't have sufficient disk capacity to do this. You're currently at 87%, you can view usage in cpanel. Might want to ask your host why there's almost 4 gb in 'Files in hidden subdirectories'.