The conversation you and your wife had over chewing with your mouth open sounds like a conversation any parents could have, not just mother and step-father. The fact that the kids are your step-children and not your biological children seems mostly irrelevant to me. The bigger problem is that you and your wife don't agree on how to raise and discipline your children.
Your kids are at an age where, at least in my experience, they like to play one parent off another. "But Mom said I could...", "But Mom always lets us..." and "But Mom said it's ok (even though Mom never got asked at all)" are some things kids try to do to leverage the fact that there are two parents instead of one. If one of you is more permissive, that is who they will go to to ask about anything they think they might get denied. If that option isn't available (for instance, if your wife is away), then the "But Mom"s come out in an attempt to get what they want.
Find some time to sit down with your wife without the kids and talk about it. If you feel like she is undermining your authority, tell her (and find a nicer way to phrase it). Tell her why you are concerned about your relationship with the kids. Talk about the things that bother you about your kids' behavior and decide what you can live with and what needs working on. Then figure out what to do together and stick to it.
