Heat pumps act differently at different outside temperatures. I did a thread on a different forum explaining what goes on.
Heat pump efficiencies and use explained - Homesteading Today
1. Should the blower continue working after the temperature reaches the setting on the thermostat? Is there a "time" for it to continue to assure the house is uniformly heated?
In cold weather, it'll remain on until the auxiliary heat has cooled to a safe preset temperature. Otherwise it remains on until the coil is about 80 (usually). The settings are to grab the last bits of useful heat.
2. What temperature should the air be that is flowing from the floor vents? I believe it is around 65ish (it feels cool).
Depends. In general, take a thermometer and make sure it is at least one or two degrees above room temperature. If your ducts are long or poorly insulated, temps can drop below that during cold weather.
3. Should I be concerned the blower is running so long so often?
In cold weather, that is the way they work, to avoid using the auxiliary heat as much as possible.
There comes a point on the coldest days that you may save money by using space heaters. See the link for more details.