I live in Texas, at the very tip of the state (Brownsville).
Language, I don't know exactly which cities you'll be, but be prepare to hear many languages. I mean we all (almost all) speak English but you'll hear a lot of Spanish too; because I live in a border city (with Mexico) I speak mostly Spanish, but being fluent in both languages, I switch to either depending of the situation. Don't worry about it. I think people will like your accent.
Guns, I've only seen cops wearing their guns, the majority of people that carrying a gun, conceal it. You don't even notice if someone is carrying a gun or not.
Tipping, in restaurants is 10-20% depending on the service, and we usually leave the money on the table or add it to the ticket if paying with credit card. I've never seen people doing the secret handshake. I would tip at restaurants, hotels, the attendant at the car wash or the guy at valet parking. If I remember more I'll come back and tell you, but I think that's it.
About the hats, not everybody wears them, boots are more common. My dad, for example, is a boots guy, and he has hats, but wears them only on special occasions.
Last but not least, I echo the suggestion of going to Hill Country, Fredericksburg has some nice cabins and the downtown has a very nice German feel to it. Wineries here in Texas are very good too, and I've been to a few around that area. In the spring you can see the famous bluebonnet blooming everywhere.
I hope I was of help, and have a great trip.