Pull down for 74HCTXX series logic ICs
In general, you should never leave digital inputs floating, unless instructed otherwise by the chip manufacturer or designer. Depending on the logic family, a floating input could be interpreted as a high, a low, or undefined value and could even lead to damage to the device by passing too much current through having push-pull transistor pairs both turning on.

If you are feeding inputs from other push-pull outputs, assuming the voltage levels are comparable), you should be fine.

Open inputs must be tied to either Vcc or ground, either directly or through a resistor.

If you have a single-throw switch connected, yes, you must use a pull-up or down resistor, depending if the switch connects to ground or Vcc, respectfully. The resistor should be low enough resistance to avoid noise injection to the input, but high enough that the current flow when the switch is closed is not excessive. 4.7K and 10K values are common.