The original barrel for the K1 was made by Lothar Walther. It was/is top quality barrel, which could shoot pellets very well. However, given the particularity of the K1's design an the fact that it shoots straight from the magazine (which acts as the chamber), this meant that by using a more "generic" (as in, not specifically designed for the K1) barrel, it required some experimentation with several pellets/slugs to finally find the ones that worked well with it. Once the user found out this, the K1 with the Lothar Walther would be a dream to shoot. But it remained pellet picky - with slugs it was so that Huben made their own specific slugs for the K1, which were also very good with the LW barrel.
In the quest to improve the K1, Huben decided to go for a barrel specifically designed for the K1, which contemplated the way it shot (from the mag) and to make it suitable both for pellets and slugs. Apart from being designed to shoot from the magazine, the other features of the new Huben proprietary barrel combine are its lack of choke, making it suitable for slugs, with a slow twist rate that would prevent pellets from over spinning while using the enormous amount of power that the K1 produces. Bottomline, it is now not pellet picky, and can shoot slugs very well. A caveat with slugs: they are always, with any airgun and barrel, pickier than pellets, and require more testing until the appropriate ones are found.

Broadly speaking, there are two K1 versions. The early one, which was produced until early 2020, and the late version, what we generally call the "2020" or "post-2020 version"; we don't use it officially but MK1 and MK2, if there are only two versions, would be correct. The major change that occurred in 2020 is related to the new barrel, which is unchoked and designed to fit perfectly with the K1 that fires directly from the magazine, and now can shoot accurately both pellets and slugs. The latest K1 also performs very well at any power level, while the original version performed considerably better at high-power ones.
Other than that Huben is a manufacturer which constantly improves quality control and listens to its customer’s feedback to improve its platform. That means that there might be slight differences between each batch, adding incremental improvements and better overall quality.
Any other labeling is the product of users, YouTubers, and/or reviewers, however, it is not officially endorsed, as they only reflect these minor changes which do not make for stark changes in parts and/or functionality.

Both the standard and the special edition share at its core the same platform. All the internals, barrel, reservoir, gauges, etc.. are exactly the same and they should perform the same. The main difference lies with the stock, as the special edition comes with a high-quality wooden stock instead of the synthetic one. Another major difference is that both the stock and the shroud are made of carbon fiber in the special edition. These three elements make the Special Edition a bit lighter, which is rather counterintuitive given the wooden stock! Another minor difference is that the shroud vents are now towards the back of the shroud and closer to the shooter. This means that you might feel the K1 rather louder (as the air would vent out closer to you), but in reality, from the front of the airgun, the Special Edition is quieter. NOTE: late Standard edition also have the vents towards the back.

Being hammerless means that the trigger acts directly on the sear of the valve with no hammer strike. The valve, driven by a piston, is immediately opened by the pressure of the plenum.
Not needing a hammer to overcome the resistance of the valve means that there are fewer moving parts, fewer vibrations (which would impact accuracy), more efficiency, and much more power. This action also makes the K1 very reliable, and indexes without issue. Accuracy - Power - Reliability are the direct benefits of this unique action which Huben pioneered.

Definitely yes. While shooting straight from the magazine could present some challenges for this, the latest K1 handles it perfectly. However, as with any rifle, shooting slugs require more testing to find out which ones work best with your particular airgun. In case you don't have that much time to try out lots of slugs, we will narrow it down to those options that we know for sure that work. In .22 cal, from Zan both options in .219 (22 and 28 grains) match perfectly, and from EDgun (and JSB) the .216 Knockout Slugs MKII work well too. For the .25, Zan delivers with his .253 slugs both in 33 and 38 grains, and EDgun (JSB) Knockout slugs .250 MKII are good to go.
Now it is up to you to narrow it down which ones work best with your own K1!