Floating inputs drain power like PacMan eats dots. Unused pins need to be configured as INPUTS with a PULLUP or PULLDOWN resistor or they need to be configured as OUTPUTS set LOW. If you do nothing, it's very likely the pins will float.
Pins Configured as InputsArduino (Atmega) pins configured as INPUT with pinMode() are said to be in a high-impedance state. One way of explaining this is that pins configured as INPUT make extremely small demands on the circuit that they are sampling, say equivalent to a series resistor of 100 Megohms in front of the pin. This makes them useful for reading a sensor, but not powering an LED.
As for the stabilizing cap.. There is no need when run 100% battery power. Your batteries are not fluctuating wildly like an AC adapter. The caps that are built in to the Arduino on the 5V rail are more than adequate. You could desolder all of the electrolytics and not notice when running on a battery
Enabling them will ensure a constant value, but this really shouldn't be a concern.
Are you sure that there is such a thing as PULLDOWN built into the standard arduino boards?