Brandon was answering a direct question from a reporter about whether police have verified his whereabouts in Mississippi on the day of the murder. That's why he mentioned the rental car and the plates, it was an answer to a question. As for his father's demeanor, Some people laugh when they're nervous. You nor I nor most people on the planet who were just normal countryfolk get put in the position of the whole world looking at them, and handle it like a pro. I would argue that the oddness and awkwardness is more indicative of innocence than it is of guilt
And again, as has been noted many times, RB was not even in the state of Texas when the crime occurred, being about 1600 miles away. So the idea that RB had an MB-crime-stained shirt is almost funny in its ludicrousness, because of the degree of absurdity required to assert he could have participated in a crime from 1500 miles away. It's just silly.