Yes, closer to "very little."
Rex called TO's. But the bulk of the offensive responsibility was on Roman and Lynn. And while it's easy to make the lazy "inexperienced" remark about Lynn, his performance as OC helped him get a HC position.
Rex hiring Roman doesn't mean he "coached" TT.
Rex Ryan's defenses have always been risky, attacking defenses designed to either get a sack/TO or give up a play. If you want to argue that his defensive personality seeped into the O, I'd like to know why you think so.
It's not like Roman's system was way different before he came to Buffalo. He was always running a complex run heavy offense that controlled the clock and barely passed. He ran the exact same offense in San Fran. There was virtually no difference with Rex Ryan or Jim Harbaugh.