1) Essentially, an AM directional array works by placing multiple towers in a certain pattern to produce a desired signal.
I'll use WBAA-AM, the radio station licensed to Purdue University
They utilize an offset
three tower array at night to produce a
signal null to the south.
This is accomplished by splitting the signal from the transmitter output and directing a different portion of it to each tower. In this array, I suspect the middle tower does most of the work. The southernmost tower would receive less power, and it would help generate this southern null. The northernmost tower would also help cancel power and generate the slight northeastern rotation the pattern has.
The more towers in an array, the more complicated the pattern, generally. A one tower array would have a theoretically circular pattern. A two tower array will typically have a carotid pattern.
2) You may know that any wire has a resistance and inductance, including the wire that stretches under the ground to provide the signal to the multiple towers of a directional array. Inductance changes the phase of a signal. Using the WBAA example again, the transmitter plant is by the middle tower. If there were no phasor, the north and south towers would be a few milliseconds out of phase with the central tower, leading to the directional array not working properly (generating a poor signal, and/or not in pattern). That is, a phasor will compensate for the imperfections of the wire in use.
3) Beam tilt can be used to focus an FM signal on one particular area. For example, the Denver area FMs probably aren't getting on the western side of the Rockies, so a couple degrees of beam tilt might be used to direct more of the signal down into the Denver metro. (I don't know if any of the Denver stations do this, but its theoretically possible)
4) An FM station can be directional. In fact, most FM stations are directional

To create an FM directional antenna is roughly the same concept as the AM directional antenna. If you have a multi-bay FM antenna, you would alter the power going into one or more of the bays. You could also alter the alignment of the bays. I say most FM stations are directional because they are often side-mounted on towers, and the tower can affect the signal propagation.