Pull-up and Pull-downs are normally used to ensure a line has a defined state while not actively driven. They are used on inputs to prevent floating lines, rapidly switching between high and low and a middle "undefined" region. Outputs normally do not need them.
But most mcu pins are GPIO, and sometimes on startup are defined as inputs instead of outputs. As you said, sometimes you don't want an IC pin input floating on startup, especially like a reset pin that you would normally drive with your microcontroller's GPIO.
This is when you use a Weak Pull-up or Pull-down on the line. Because they are weak, and you choose the default state, they provide no interference with your circuit (If the input is supposed to be low at all times, then pulled high, you choose a weak pull-down, and vis versa), but they do consume a bit of current. This is why you choose a resistor weak (Higher the value, the weaker) enough for the job.
Another normal output setup that uses pull-ups (or pull-downs, rarer) is Open Drain or Open Collector connections. These only drive a connection low, or release the line, leaving it floating. The pull-ups are used to bring the line to a high logic state.
Best Answer
There is no good reason to wire a capacitor directly across an output. If, for example, the capacitor is discharged then it behaves as a momentary short-circuit when the output is turned on. This presents a momentary overload to the output. Similarly when charged, pulling it low will effectively present a momentary connection to V+.
On the other hand, as an input R and C behave as a low-pass filter. This could be used to debounce a digital input or stabilise an analog reading. In both cases it introduces a small delay in the response time.