I have Magento version 18.104.22.168. I installed a custom theme, one calendar and another calendar. The new calendar company had to tweak it a bit and somewhere along the line - things got messed up (like - I cannot display the contents of my categories in Admin and therefore cannot edit them or add new categories). I was thinking - is it possible to do a fresh install of 22.214.171.124 on the same VPS server and just connect the 126.96.36.199 database from the existing installation to the new 188.8.131.52 install? I appreciate any suggestions.
You can install the newest version of Magento but you database will be upgraded. I would recommend that you try to keep up-to-date with all your versions.
For example: Once you point your new version of Magento to you 1.9.1 database, your database will automatically be upgraded to 1.9.2 (You can not go back on that database so make a backup!)
You should maintain all your installs with the 1.9.2 version.