Instead of having the players cover every single room in a dungeon or square of travel overland, is it permissible (or even warranted) to "fast forward" to a more interesting place, provided that there's nothing interesting going on between points A and B?
In any circumstance where you're trying to shoo in a sense of urgency and you need to be at the castle to rescue the princess as soon as possible, and you're sure that the princess is actually at said castle after performing divinations or using your information and contacts to confirm her location, fast forwarding keeps the emotions at the table high and lets you ride out that climactic feeling that being in a hurry is.
In situations where you're also above whatever level you were planning for your campaign, fast travel also lets you remain at a consistent level without having to roll on any kind of random encounter tables that could push your PCs into the next level by chance.
However, in situations where you're in no particular rush and you have an event planned on the way to the castle and you aren't in a situation where there are time sensitive things happening, by all means take the scenic route, get wrapped up in a side quest, fall in love with a Dryad, kill a giant troll, or finish whatever activity you had planned.
I do this all the time in my campaigns. I play with my group during the school year since we are in college. So running my game is more of a Cinematic feel. If they need to travel 200 miles from one town to the outskirts of where the adventure truly lies, I deem it unnecessary to "waste" our limited game time on random bandit attacks on the road. That said, I also grant my players levels after important plot points have been completed and not based on XP. If I was giving xp I might throw a few random encounters at them if they need a little extra help staying at an appropriate level for the encounters in the campaign. The one exception to my "skip the boring stuff" style is if I want to highlight the expanse of the journey they need to take to get from point A to point B, but in that case I am forcing them to travel for a thematic reason.
Passing time is a useful technique. Time should always be passed when no one is interested in a given period of time and no result of the time period will matter later. Time should not be passed if anyone is interested in the events of the time period (usually because they want to act in it) nor should it be passed if any player's understanding of what is going to happen in the passed time is wrong; in either case time should be skipped up to the point at which a player wants to act or an understanding is revealed to be wrong. Eventless journeys are often uninteresting, and so are usually good to skip to the end of, unless your group likes rping campfire scenes or the GM (or other person with the capacity to do so) wants to narrate some scenery.
I do stuff like this for things like retreading through cleared dungeon floors, but for stuff like exploring the dungeon itself it's kind of expected that you at least give them the ability to search through the map and try out all the different paths themselves- if you just strip away the map and skip right into the necessary encounters one after another then you kind of lose half the game.
Yes. Just tell them the story about what happens, taking their character's normal behaviour into account, and only play the interesting parts. Better is to do the same but allow them input and ask for clarification as you tell them and just decide what the results of their actions are to fit the story.