If there are specific things you (or your mother) want to talk about but not in front of your cousin, I think it would be reasonable to ask your aunt "Can we go someplace else for a bit? There's something I want to talk about that I don't think is appropriate for Billy." That gets away from a visitor asking a child to leave the room in his own home--and your aunt, who does live there, could tell him to go somewhere else for a bit in the house for a bit. If it's more that conversations drift to such topics but only stay there for a little while, consult the child's parents (who would be your aunt and uncle, if I have this right), about topics they don't want him to hear about, and how to deal with it if he's in the room. This would of course be trickier if you, your mother, and your aunt consider some topics inappropriate for a ten-year-old boy but that his parents think would be fine for him to hear about.
With regard to seating, I think the boy is a red herring. If neither you nor your mother is comfortable sitting on the loveseat because of dirt, it wouldn't help much for Billy to be somewhere else: in his absence, one of you could use the computer chair, but the other would be either standing, or on the loveseat.
Also, while this goes beyond etiquette, you clearly don't like this child, and it sounds like you're trying to be big-sisterly out of a sense of duty, not love. Almost nobody suffers from the lack of someone to fill the role of annoying little brother, and it might be better to drop back from that to different-aged cousins who don't have much in common, or hang out together except by chance.