You can use the Romex inside the house. That part is OK.
Legally the ground rod has no significance. It might help establish your ground plane in a practical sense but the code is silent on it. If I was doing that I would add a #8 ring around the pad. I am not sure about the rubber mat thing. That may be a local accommodation but I am not sure where they justify it in the code. I have seen 3 foot or wider decks around a spa going for the same thing and that sounds better to me. A wet rubber mat sitting on an energized concrete pad would be energized.
You need to get a ruling from your AHJ on this part. I am not sure what a totally legal solution would be without chipping concrete. (Maybe saw cut a groove around the spa, push in a #8 and pack it with grout?)
Bonding conductors are not #10, they have to be #8 solid copper.
