Making SS Permanent

Is there any benefit to making a SS permanent? I don't want to waste the gold if there is not.

One major benefit is that you have those SS even through reincarnation, so when you reincarnate you don't have to start all over with brand new SS that you have to buy and re-level. Right out of the gate you get to to start story missions and use adventures to farm resources. Makes starting over alot easier and faster. I'm actually intended to make a guide on here pertaining to this very thing when I got time, sorry I haven't done this yet

Do the SS keep their levels/stats through reincarnation? Or do they start over at 0 also?

No, any SS that you make permanent should stay as they are, any SS that aren't permanent will be lost completely as well as any gear that isn't permanent before reincarnation. With SS the cheapest way to make permanent is to spend the 3 gold unlocking the 1st seal slot, and with gear the cheapest way is to spend 2 gold renaming it.