To answer your first question, it is not a normal behaviour that the routes from the device’s web interface do not appear in the routing table by viewing it with route command. I have tried to set static routes via SSH connection myself, however, I was unable to replicate your results. The routes do appear in the table in both instances. However, I would like you to check, whether you were looking for the routes by going to Network -> Routing -> Static routes or did you check Status -> Routes -> General routes in the WebUI, since the first option is used purely for interface configuration, the other is used to view what is currently set by default and already configured.
As for the backups there are a couple of options. The manually set routes are saved in the current configuration profile. The current configuration can be set as a new default configuration by going to System -> Backup -> Create default configuration. You can also create a current configuration backup file, save it and upload it later in the same page under the Backup configuration section. You can also save the current configuration profile by going to System -> Profiles -> Configuration profiles and simply apply it any time you want or made any changes later on. In case your routes disappear after system reboot, you can add commands to create static routes during system boot in /etc/rc.local file.