First, is the service you have with hosting provider on a shared system? And, if it's not an international secret, who is hosting provider (they are not all equal in support of Moodle 2+)? Most hosting providers do have forums and FAQ's. One of those FAQ's might be related to Moodle.
Am guessing shared system so it's not only the version of MySQL required for Moodle 2, but, also, the resources a shared user is allowed to consume/use. Restoring any 1.9 course to any version 2 will be pretty intensive ... and might be over the limits allowed.
Before beginning anything server ... backup. Backup the code directory, the data directory, and get a MySQL dump (sql file) of your database. Hosting providers usually provide different tools to accomplish the above. Download your backup to local machine. Again ... before beginning server, BACKUP!
What might be the first step is to upgrade the 1.9.5 to the highest/most secure 1.9.19. This is not a waist of time and probably needs to be done anyway!
Linux server and uploading extracted .zip?? (which is intended for Windows) - often wondered why that was ever recommended. The .zip file is larger than the .tar.gz - which means there are more files. The tar.gz is packaged for Linux/MacOSX servers - which happens to be what you are using on the remote system.
Basically, when upgrading, one BACKS UP, then keeps the data directory and the config.php file in the current install (the database isn't touched, yet ... cept to backup). Uploading all the tar.gz extracted files to the moodle code directory, setting appropriate permissions/ownerships on all files/folders uploaded, one then logs onto the Moodle as an admin level user and the process to move 1.9.x to the highest 1.9.x should begin.
See: http://docs.moodle.org/19/en/Upgrading
If your current Moodle is in a 'moodle' directory? You could install a 2.2.6+ in a moodle2 directory AND try the backup from 1.9.x restore to 2.2.6+ on the server ... where it has to work.
'spirit of sharing', Ken