I got a tetanus shot at the ER after stitches, is this okay because I had a tetanus shot 6 years ago.?

Yes. There is some data supporting routine tetanus innoculation with a deep wound after 5 years. An extra tetanus, at present, does not appear to cause any problems.
It's fine. If you have a cut requiring sutures, most doctors will give a tetanus shot if you haven't had one in the last 5 years. Better safe than sorry!
Yes. It is appropriate to get a tetanus shot if you have a deep cut and it has been more than 5 years since your last tetanus shot.