What does the fact that Tom conducts his affair in public say about him as a person in The Great Gatsby?
In The Great Gatsby, Tom likes to show his wealth off. I'm answering your question based on how it originally appeared, when it read "affairs" instead of "affair." I take "affairs" to mean finacial goings on, like flaunting his polo ponies. The fact that he flaunts his wealth reveals his character. He is on the top of the food chain and likes people to know it. He sees himself as superior to others and is a part of the bourgeois, middle-class status quo.
Your original question also contained a question about the green light losing value to Gatsby once Gatsby had Daisy. The green light on Daisy's dock is a symbol of what Gatsby doesn't have, and of what he wants. It's his dream. It's what he longs for. That's why he stares at it. It's also a sign of how close Gatsby is to finally achieving his dream. He can actually see it across the bay.
If you're correct about the green light losing its significance, it would do so because Gatsby has the real thing, at least in part. Once he has Daisy, or thinks he does, he doesn't need to stare at the light anymore. He has what the light represents.
I think Tom's affair with Myrtle being staged in public says something about his character and integrity.
First, I think it shows great disregard for his wife. He doesn't host this affair in front of her face, but she seems to know because of the calls he takes and the way other people know.
Second, it shows a great arrogance. It's like he knows he won't get caught, or even if he does that it won't hurt his marriage any more than it already is hurt.
A person who will act this way is disrespectful, apathetic and careless to others.
Pretty much everything we see about Tom Buchanan makes him look bad, doesn't it? The fact that he carries on his affair in public is just one of those things that makes him look bad.
We can see from this that he is very arrogant. And we can see that he does not really care if he hurts his wife (or Myrtle's husband). So I guess that makes him selfish too.
This sort of goes along with his racist ideas -- he thinks he's better than other people and that rules don't really apply to him.