I recently had some interview footage shot by a local public TV station, and to me the sound is flat and boxy. Would using an overhead directional mike make the sound more full?
Sound is a multi dimensional discipline. You always want the microphone as close to the subject as you can get it, either as a lapel mic or boom (over head) mic. You could also mount a mic on a stand to the side, just out of shot.
If you record using the built in mic on the camera you are actually capturing the sound as it's bouncing off the walls - hence the boxy feel to it.
Yes it will, as long as you can put it in position.
Some lapel mics are unidirectional, others are omnidirectional. It makes a difference.
However, the reverb in the room can make a huge difference, so the closer you can get the mic to the subject (and reduce the input levels) the better you will be at isolating the voice from the room.
Unidirectional mics pick up (for the most part) only what you want 'em to because they only pick up sounds in its sweet spot, which you can aim at your subject. Omnidirectional pics up all ambient sound (including boxy-sounding wall bounce).
IMO, for interviews, unidirectional mics offer clearer audio. You pick up less florescent hum, air circulation noise, producer's paper shuffles, etc.