Whether a 3.3 V signal can reliably drive a 5 V logic input depends on the minimum guaranteed logic high level of the 5 V logic input. Whether that is sufficiently below 3.3 V for reliable operation varies with logic families and manufacturers, so consulting the datasheet is essential. For example, Microchip PICs with Schmitt trigger inputs usually require 80% Vdd for guaranteed logic high. That would be 4.0 V, which is too high to be driven from 3.3 V logic.
A easy hack is to use logic with "T" in its name, like 74HCT as apposed to 74HC. The T stands for TTL, and means the logic input levels are compatible with the old TTL logic. This had a finite guaranteed logic high threshold sufficiently below 3.3 V so that driving one of these from 3.3 V logic is fine, although its a good idea to check the specs in any one instant. So if you need a logic gate in there anyway, make it a HCT type and you get the 3.3 V to 5 V conversion for free.
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.
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It may be required to pull the output up if/when the inputs are 'floating' (open-circuit). To understand its function we would need to know more about the circuit.
Whether the logic level is indeterminate depends on the IC. If you have its datasheet then you can find out what voltages are defined as 'high' and 'low'. Otherwise you just have to trust that the circuit has been properly designed.
I suspect that the inputs are not actually voltages, but Open Collector signals that pull down to ground when active. If this is the case then the output voltage will either be Vdd or close to 0, with no 'indeterminate' middle voltage.