May Not Matter. There is an old study that demonstrated cool air/moisture penetrates deeper into the respiratory tree. But not by much. There may be very little practical difference.
Cool Mist. If you choose to use a humidifier, cool mist is preferred. The heating of the water in a warm mist humidifier promotes mold and bacterial growth that can have adverse effects.
Cool mist is safer. In general (with a few exceptions) it is more about raising the moisture level in the room. This is especially useful in winter months when the air is quite dry (at least in cold climates) and little noses and throats tend to be dry and irritated from all the colds going around. The real advantage to cool mist is that curious toddlers won't burn themselves when they grab the humidifier.
Depends on local. In the north, where humidity is often handles passively by a furnace add on, these have little use. In the south, where dry heat is the rule, children and adults may become symptomatic during long cool spells. Headaches, sore throats, nose bleeds & dry coughs are often related to heater related humidities in the 10-20% range.Babies are very sensitive to these issues.I recomend them.
No. It just is not necessary, although it is useful in croup.