1. Is it plausible that a character in a medieval setting, knowing nothing about modern-day science, will start building something like that?
-Not categorically, no. But the science of using barrels to stabilize and aim shots is already established. To my knowledge, there are a couple of examples that use this principle.
On page 42 of AA ("Aventurisches Arsenal"), you can find the "Gandrasch-Armbrust", which is essentially a crossbow that, instead of using a lay-on rail, uses a tube in which the bolt is stored and accelerated,
Furthermore, there is a whole class of almost gunpowder-like weapons. They don't use a chemical though, but torsion power.
These weapons are the "Arbalette", "Arbalone", "Balestra" and "Balestrina", you can find them in AA.40-41 and AA.43. It is stated on the latter page that Hunting-"Arbalettes" as well as the small pistol-like "Balestrina" are already widely used by civilians, albeit I guess this is only true for nobility.
These kind of weapons shoot (at least in their common civilian variants) lead balls, regulated weapon models especially for the military also may come as bolt-shooters with or without fast-loading mechanisms or magazines. Regardless of the model, all of these weapons use the talent Armbrust (ger. Crossbow).
I would say that if at least one character has seen or inspected such a weapon may at least get the idea of testing something like that.
2. If yes: Would you include any skill checks for it? There are some things that might fit in TDE, but they are usually so unbearably useless that no-one ever skills them (Which might be a good reason to ask for skill checks on them)
For the idea? No skill check necessary. For building it? yes, absolutely.
While coming up with the idea is plausible even without proper founded knowledge of the character, actually building their creation is a different story. There are many things that come into play when building a weapon.
In this case, the placement of the tube has a huge impact on the quality of the weapon. While there is no complicated mechanism to propell the projectiles as it is the case with mundane weapons like the Arbalette, it is of importance how precise the placement of the tube will be, as it impacts the ability to aim and the probability of a failure of the weapon through misplaced force vectors on the projectiles, risking the destruction or deformation of the tube. This could also lead to projectiles hitting things way out of aim, as parts of the projectile ricochet from the barrel inside near the muzzle.
As such, I would ask for the following checks:
If the tube needs to be built (and not bought): Depending on the method "Grobschmied" or "Metallguss"
To create the leather straps or ropes to affix the tube: "Lederarbeiten" or "Seilerei" if not bought
To fix the tube to the "Hammer of Magus": "Mechanik"
To shoot, use "Armbrust", or at least MU+IN+GE / 3.
These crafting checks should have influence on damage, range, loading time and Patzer chance, depending on how good they were.
3. If you allow it, how do you balance it against existing game items and mechanics?
As said before, Arbalettes shoot lead balls. These are likely cast into shape and will have a much better shape than rock and stones ever will. Lead is also more stable and is less likely to splinter in every direction. It is easier to load and far less likely to jam in the tube and explode it when the "Hammer of Magus" forces them out of the barrel while the stones are wedged into each other. Since the shape of rock is so irregular, loading will still be faster than the 30 or so actions needed to load an Arbalette (civilian), but will be longer than loading a Balestrina (e.g. 2 Actions). Stones may hit hard, as we know from slings, but they dont really do wounds, so I would say have it lead to a wound at KO instead of KO/2.
It has quite some force, but I recon it would not hit a single target with more than 3 stones, I would say it has a damage of 3W -3 plus a bonus you as the GM decide due to the TP* of the "Hammer of Magus".
However, since the weapon uses improvised projectiles (and basically is an improvised weapon almost), I would have the Patzer ("gaffe"?) probability set to be much higher than normal weapons. You could invent a Patzer-Tabelle for this specific idea.
Also, since stones aren't exactly the most aerodynamically stable projectiles (and the wide barrel for loading more stones also reduces range), have a range table like this: 2/4/7/10/15
As it comes to mind, the force of the "Hammer of Magus" may have enough force to, if the right/wrong stones are loaded, shatter the load and only shoot small splinters. You could then say that RS (armor) against the shot is doubled.