I currently have two direct TV boxes at home, but want to connect a third TV. Can this be accomplished by connecting a splitter to the primary incoming cable line that goes into the DTV box and then to the third TV?
No. If you have a dual tuner receiver you can connect a second TV, or you can use whichever outputs aren't being used if it's a single tuner, HDMI, Component, but you'll have to watch the same thing as the other TV if it's a single tuner receiver.