Disc game have certain firmware requirement, so as long you have the required firmware it needs you should be fine being offline, just that you can't play online.
You can't if you want to be able to use downgrade in the long run, the lower the firmware the better you are when it come to these exploit and downgrade, for now the downgrade only work if you were on lower firmware.
We have no idea if the downgrader method would work on 7.51 or higher firmware, the downgrader is meant for people that is on lower firmware, you can't access the PSN without updating thought, so if you want to download you would have to update.
Disc game have certain firmware requirement, so as long you have the required firmware it needs you should be fine being offline, just that you can't play online.
If the system know there is an update available for that game the system would force you to update, so I think you might have factory reset to not get the update nag, probably get another unit if you want to do all the exploit stuffs.