Establishing first of all, Twst has kept Skully's nature vague

• Simply put: we don't know if Skully is human. In his live 2D, Skully's ears are completely covered. Even in his illustration, we can barely see his ears. It's an interesting design choice. However, in his chibi (which wasn't revealed until Episode 5), we can see his ears aren't shaped like a fae, but are round like that of a human.
• However, it's pretty hard to think he's human either, when we consider his "moshi, ne moshi" greeting in Episode 1 and its connection to the supernatural. Voiced, to be noted. Exactly what kind of creature he could be is currently unknown, but because this part was voiced, it's something we need to pay attention to at the very least.
• In regards to his mortality, the only clue we're really given is Skully himself saying that he may never meet us except through Halloween, which implies a lifespan similar to a human or something similar. But Skully doesn't know how far into the future that the cast is from. Heck, does he even know of his nature?

Now, addressing that invisible kiss...

• I feel like this is the main thing that told all of us that Skully is long gone. Heck, if you catch me in my right mind, I might just tell you that yeah, that should be enough proof that Skully is dead! Little signs from the dead such can manifest in similar ways to what Jamil and Leona sensed: hearing a kiss and feeling a kiss on your hand.
• (unfortunately i'm not in my right mind so you're getting my dumb reasons why i'm arguing for otherwise)
• However, if Skully is gone, I find it interesting that this is the way they decide to show it. We already know from the very first Halloween event that ghosts can manifest all year round in Night Raven College due to the high concentration of magical power in the area. Outside of that, they cannot be seen. If they really wanted to confirm with the audience that Skully is a ghost, they could have had him appear as such at the end, when they all returned to Night Raven College.
• But they chose a scenario where people can draw a lot of conclusions from it: Is their senses getting fucked over as they wake up? Are they still reeling from the magic of the book? (though rationally, we know that Leona is not one to be easily stunned, so the first question is at the very least easy to dismiss)
• I've also seen some people theorize that he's using invisibility magic during the invisible kiss scene. And well, while I find that funny, it makes me wonder if he could be some other kind of spirit that's not dead. Like an undead of some sort, which the residents of Halloween Town are. Heck, Azul's card line about Skully talks about how he seems to fit right at home with the Halloween Town residents. Again, we don't know Skully's true nature, but the possibility of him being a species that can turn invisible is interesting.

Moving on, we have what Dire Crowley stated about Skully

• At the end of the event, Crowley tells us that he found Skully’s portrait while rummaging through the storage and shows it to us and the 11 boys who went inside the book
• What he tells us is that Skully J. Graves is a NRC graduate from hundreds of years ago, before Crowley was Headmage. Note that he only was appointed for the position 100 years ago; the Skully we met is at least from around 200 years ago. During his NRC years, Skully got to share Halloween to NRC. It was a hit, and when he graduated, he spread Halloween all over the world in his travels.
• I'd like to take a little detour first to discuss something that's been weighing on my mind: Some people have thought that Crowley is lying to us when he speaks about Skully and his achievements, but... I don't see why he would lie. For one, there is a decent chance that Crowley might not have gotten to see Skully. Even if he did, it's even more unsure if he was involved in Skully's affairs.
• However, there's one main reason why I do think that Skully got to live a fulfilling life instead of facing an overblot that killed him or some other tragedy. I'd like to dedicate a longer post to this matter, but to make it concise, I think through Skully, Twst is starting to establish something new regarding their history. I think that historical teachings, folktales and stories, and rumors that are well-known tend to be lies or twisted truths. Whereas those that are obscure and not known are actually what occurred. Skully is called the King of Halloween who's done so much to spread the holiday to the world, yet not even NRC students, who should be the first to know considering that Skully is an alumna, know of him. Although there may be other reasons why that's the case, I like to think that at the very least, Skully's obscurity indicates that he did live the life he wanted and succeeded in working for a future that generations after him can enjoy.
• Anyway, that actually isn't the main point of this section of the post, but I kind of want to air that out first. The main point is that Crowley only really said that Skully is a former NRC student from hundreds of years ago who traveled around the world to spread Halloween. But he never told us where he was buried or whether he saw his ghost roaming NRC. He never said anything about Skully being dead.
• He probably said the hundreds of years ago bit, carrying the assumption that of course, Skully may have passed away. But we have to remember that we have long-living species in Twisted Wonderland. Fae that were students 200 years ago are very likely to be alive now. Crowley himself is a long-living creature, having been Headmage for 100 years. I think with that in mind, it's important for him to emphasize that Skully's gone if he really is. But he never mentioned it. Therefore, there is a good chance that Skully might just be somewhere else. That or Crowley just doesn't give a fuck where his alumni go, and I might be thinking too deeply about the absence of certain words. Honestly, that's a pretty good chance too.

Lastly, we have the scene where Jack Skellington gets shot down but survives it

• Here's where it gets a bit more into speculation, but you're going to have to hear me out.
• In the movie, Jack Skellington gets shot down by the military for impersonating Santa Claus. When the Halloween Townspeople watched it, they all despaired, and the mayor started declaring to all that Jack has been blown to smithereens and proclaimed him dead.
• But that's not what happened: we find Jack landing on an angel's statue, alive and definitely not blown to smithereens.

• And considering that Skully is still very much twisted from Jack Skellington... do you think the writers are pulling a similar move? Making us think that he's dead, just as the townspeople thought he was dead, only for us to learn eventually that he's alive.
• This of course depends on what happens eventually in the sequel. From the way things are proceeding, what happened in the Lost in the Book with TNBC is events prior to the movie, and we could very well have the sequel be set during the movie events. (If you ask me, I kind of doubt that actually. I feel like Skellington got inspired by Skully's love for Halloween, enough to stay true to his identity as the Pumpkin King, which would mean the movie wouldn't happen the way we know it. So if anything, I feel like TNBC 2 would focus less on the actual movie and more on the side games where Oogie takes over Halloween Town and even kidnaps Santa and other people, but I admit, that's a stretch, especially considering that we will get Santa giving Halloween Town a taste of Christmas)