Disclaimer: Possible spoilers about LMoP.

So, I have read through the rewards of the third chapter, plus the "starting point" of the two first chapters. The needed encounters to reach the recommended 4th level can be answered by math. From a point of view of experience, for 4 players, you need a total of \$ 4 \cdot 2700 = 10800 \$ XP for the party.

The first chapter gives you a total of \$1400\$ XP (350/character).

The second chapter gives you a total of \$+2800\$ XP (+700/character).

The Cragmaw castle, from the third chapter, gives you a total of \$+4400\$ XP, assuming

the party clears the castle completely, rescues Gundren and you include the optional Returning War Band encounter.

From the main story, we have a total of \$ 8600\$ XP. That means the Side Quests should give the party at least \$10800 - 8600 = 2200\$ XP. I have split the side quests in four: Agatha (sent by Sister Graele), Wyvern Tor (sent by the Mayor), Old Owl Well (sent by Daran) and Thundertree (sent by Quelline Alderleaf).

\begin{array} {|r|r|} \hline \textbf{Side Quest} & \textbf{Total Awarded XP} \\ \hline \textrm{Agatha} &200 \\ \hline \textrm{Wyvern Tor} & 1250 \\ \hline \textrm{Old Owl Well} & 1000 \\ \hline \textrm{Thundertree} & 3750 \\ \hline ~~\textrm{Westernmost Cottage} & 50\\ ~~\textrm{Blighted Cottage} & 50\\ ~~\textrm{Brown Horse} & 200 \\ ~~\textrm{Blighted Farmhouse} & 200 \\ ~~\textrm{Ruined Store} & 400 \\ ~~\textrm{Dragon's Tower} & 2000\\ ~~\textrm{Old Smith} & 100 \\ ~~\textrm{Herbalist's Shop} & 200 \\ ~~\textrm{Old Garrison} & 250 \\ ~~\textrm{Weaver's Cottage} & 150 \\ ~~\textrm{Dragon's Cultists} & 150 \\ \hline \end{array}

Essentially, the side quest giving the most XP is, by far, the Thundertree, due to the Dragon encounter. It is also the encounter that gives the most valuable magic items - +1 Weapon, 2 scrolls and 2 potions. Old Owl Well also gives a Ring of Protection, which is important.

From that, simply don't take Thundertree from the party, which is also the safest way to learn about the Castle's Location through Reidoth. That doesn't even account for the random encounters.

As a second option, the Cragmaw Castle can be skipped, actually, as long as the party can get \$400\$ XP from the random encounters, which shouldn't be hard.

As a side note, you shouldn't be worried about the PCs being able to find the Castle. The Goblins and Hobgoblins from random encounters are able to provide that info, so the party only needs to wander around for some time.

I have not played nor DMed SKT. I have some experience with sandboxes, though, both homebrew and published. For published sandboxes, I would say most of my DMing time was with Curse of Strahd, so my advices might be a little off for SKT. I hope we get better campaign-specific answers.

For actually learning the content needed for a session from a published adventure, I would usually take roughly the same time as the session, e.g. 4 hours preparation for 4 hours session.

With your time restraint, it seems more like a 8 hour session for 2 hours preparation. From my experience, that is simply not enough time to learn what you need. It means you will need to learn it with a lot of antecedence. For that, some things I usually already do, but are even more important for you.

"Study" the campaign

That means read the whole campaign before starting play. Use whatever methods you usually use for learning things - make notes, simulate yourself speaking as an NPC, tell the story of the campaign to someone else that is not going to play.

By actually learning the campaign before the time, when you play it and have the 1-2 hours for "preparation" you mentioned, you can just review what you already know - remember some details and all that. You won't actually need to learn that again, just like you don't need to learn the combat rules because you already know them.

In particular, make sure you know well the main story. As you mention yourself, you want to stick to the main story arc. Then focus on learning that.

When I asked about strictly following the book, I was mostly thinking about NPCs, side quests and locations. It seems you major concern about following the book is with following the main story, so, about everything else

Don't worry too much.

If an NPC is described in the books as the funny, joking one, and then you forget it and play him as the grumpy and sad one, it is fine (unless he was like the Court Clown). If a location said the market is in the center of the city and you placed it in the east corner, it is fine (unless that distance is mechanically important somehow). If you should have given that important piece of information and you forgot... it might be fine. Worst case, players lose some time, hopefully still having fun.

Remember: the main goal of TTRPGs is to have fun, you and everyone else. This is a general concept that applies to essentially every RPG problem. Even though I told you to "study" the campaign, you are not playing it to pass a test. It's fine if you make mistakes. Your players probably will make mistakes as well. Have fun.

About my third question: the thing is, while loots and encounters are easy to prepare (as you yourself said and I agree), they are the things that actually need preparing. Creating a balanced encounter on the run without breaking completely the pace is a nearly impossible task. Make sure you have those actually prepared.

NPC personalities, locations and everything else are easier to improvise. You mention you don't feel secure for that. Well, first, read the section above. Second, read the first section. It won't be a problem when they want to go to the next large city because you already read about the next large city and you know the important locations and NPCs that will be there. You probably won't remember if that was the Stonehill Inn or the Blue Water Inn, but that's fine. You know there is an Inn with an important NPC that will give an important quest.

It might get overwhelming

As I said, I have not played SKT, but I do read the internet. Perkins puts the campaign at 100s of hours, Mearls concurs with 20-25 sessions of 6 hours for a total of 120-150 hours, and the Internet often states similar times of 100 to 200 hours.

That might be too much for you to learn and remember. In a week, you probably won't even finish the campaign, even if you play 8 hours a day for 7 days, that's still half the stated expected time. That means you don't need to actually learn everything, in particular the end-game.

Unfortunately, without having played the specific campaign, I can't offer further advice on exactly what chapters/sections are the most important for you, and what chapters are likely to not get played in your gaming week.

By the way, my question about playing another campaign was just to make sure. In particular, I would personally suggest playing something that you could finish during that week, i.e., a ~50 hours campaign. That's not a too important suggestion, though.