The man sitting closest to Coulter was Matt Drudge (Billy Eichner; I looked him up while watching, and his picture matched the character). Based on the cast list, the other man must be George Conway (George Salazar), whom Coulter was dating at the time.
I've been in West Wing offices, and they are not anywhere near as spacious and well appointed as what is being shown. It's more like being on a ship in tight quarters. And, yes, the lighting is absurdly dim.