The FC-1T-1VAC has a rating of 7 FLA at 120/230 VAC for a PSC motor. PSC stands for Permanent Split Capacitor. So yes, the FC-1T-1VAC can operate capacitive start/run motors.
In Shut-off mode, the control shuts off the fan when the temperature is below the set point. In Idle mode, the fan runs at the idle speed when the temperature is below the set point.
No. The fan will speed up from either a full stop (shut-off mode) or whatever the idle speed is (idle mode) over a 6°F range until it reaches full speed.
The limiting factor is the variable stage rating. Add up the current draw of the fans you want to control and compare it to the variable stage rating. If the combined fans draw less than the variable stage rating then you can control them.