Thanks for the replies.
1) Based on the 3-way wiring
, it sounds like the switch that has the white wire is the hot circuit feed and the other 3-way with the black wire terminated to the black screw should be the switched loop wire that extends out to the ceiling boxes. This needs to be confirmed with the volt meter.
2) Set the meter to the AC setting and the 200 volt range. Leave the two 3-ways backed out from the box, but don't disconnect any wires. At the 3-way with the white wire, place 1 meter probe to the white wire screw terminal and the other meter probe to a bare copper ground wire. Now toggle both 3-ways ON and OFF, you should measure a constant 120 volts on the white wire. If so, this switch is the constant hot circuit feed and the other 3-way switch that has the black wire is the switched loop wire.
You can also confirm the opposite at the 3-way switch with the black wire. Repeat the same voltage test. Toggle both 3-ways ON/OFF. If you measure 120 volts on the 3 way with the black wire and the voltage goes from 120 volts to 0 volts when toggling ON to OFF, then this confirms that the black wire is the switched loop wire.
3) At the box that contains the black, red, and white, were the black and red spliced together using the plastic wire connector? Is this box where the fan is installed or is this box the old fluorescent light?
Any idea where the red wire terminates to at the other end? Possibly inside the fan box or is for some downstream receptacles?