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.