It may help to know that mining was a fairly stratified activity in medieval Europe. The status and conditions of the miners varied considerably depending on what they were mining. At the top of the tree, in terms of status, were the men who worked the deep silver and gold mines, which involved tunnelling to depths of up to several hundred metres in some cases. These workers were skilled men, earning relatively high pay and benefitting from a range of legal protections, including the right to take wood from private property. In some parts of Germany they also enjoyed access to special courts, where cases that involved them would be heard on favourable terms.

At the other end of the scale were the men who worked the open-cast mines. These were in a significant majority at this time. Most mining in medieval Europe was open-cast, and most of the work – for example the extraction of coal – was done by semi-skilled labour using readily available tools.

Salt mine workers seem to have fitted in somewhere between these extremes. Unfortunately, very little seems to have been written on the lives and times of medieval miners of any sort, certainly not in such an early period. Thus for example Hoffman states that

the weak understanding that historians now have of medieval mining is because it was hardly ever described in writing, and the rare exceptions were by authors who knew little about it.

Moreover, while there is evidence of salt mining in the medieval period in some parts of southern Germany, especially the Tyrol, the main salt mines that were worked in Europe at this time were in western Poland, around Wroclaw and Krakow, and in Wallachia and Transylvania. The best description of early salt mines that I'm familiar with actually comes from Ocna Mare, in what was then Wallachia and is now Romania. It is by Laurențiu Rădvan, and concerns research on the 15th-17th centuries. Given the difficulty of locating something more specific to your query, I hope you won't mind me quoting from that.

Eight kilometres south-west of Râmnic, the town of Ocna Mare developed. It stood near one of the largest salt mines in Romanian-inhabited areas, the largest in Wallachia. The place where salt is extracted is called an "ocna" in Romanian... from the 16th century on, "mare" (great) is attached to it.

Ocna Mare became a town in the 15th century. There are signs that the settlement had gone beyond târg status ever since the reign of Mircea the Old (around 1402-18)... A 1502 document tells us there were at least two priests in the settlement. Finally, the presence of a tailor in a 1516 document confirms that there was a town near the salt mine.

The salt mines were seen as the property of the prince, therefore, the settlement here was on his domain and had received a privilege... Their organization and status separated the salt mines of Ocna Mare from neighbouring Râmnic, which they were not reliant on. The ruler charged an official with managing the salt operations. He was called the salt mines chamberlain... It is only from the 16th-17th centuries that information on the work system in the salt mines is available. Along with the salt chamberlain, townspeople had the right to bring workforce in. In one trial, the town pârgari [town councillors] are accused of bringing in people from one monastery's village to work illegally. They were empowered only to bring to work the serf peasants in villages around the salt mine.

The salt workers were divided into saltcutters (who would cut the lumps of salt inside the shafts) and master lumpers (they pulled the lumps out and transported them to storage). Many documents mention the Gypsies of some monasteries who were exempted from working in the salt mine. But not all Gypsies were exempt: the gold-digging Gypsies of the Cozia monastery were not entitled to this liberty. Most saltcutters were Gypsies, while serfs were tasked with pulling out the salt. Those under punishment with the ruler also ended up there.

Hopefully this gives you a few ideas for setting up your RPG. As an added bit of colour, albeit one slightly late for the period you have set your heart on, and rather far from Germany, you might like to know that the Prince of Wallachia for much of the period between 1448 and 1477 – and hence the proprietor and chief beneficiary of the local salt mines – was none other than the notorious Vlad III Tepes, better known as Vlad the Impaler, who was nicknamed Dracula and is, of course, often supposed to have been one of the models for the Bram Stoker character.

Finally, if you are more concerned about starting your story in the most unpleasant environment possible, and are either ingenious, or less worried about the logical constraints of your game, I would recommend considering re-setting your opening in the salt mines operated by slaves in the southern stretches of the Sahara Desert, which were much closer to the hell-on-earth setting you seem to be envisioning.

Sources

For European mining in the middle ages

Richard Hoffman, An Environmental History of Medieval Europe (2014)

Laurențiu Rădvan, At Europe's Borders: Medieval Towns in the Romanian Principalities (2010)

Martin Stefanik, "The Kremnica Town Book of Accounts: The Economy of a Mining and Mint Town in the Kingdom of Hungary," in Roman Zaorel (ed.) Money and Finance in Central Europe during the Later Middle Ages (2015)

Jeffrey R. Wigelsworth, Science and Technology in Medieval European Life (2006)

On Saharan salt mines

John Hunwick, Timbuktu and the Songhay Empire: Al-Sa’di’s Ta’rikh al-Sudan down to 1613 and Other Contemporary Documents (1999)

Lasiné Kaba. “Archers, musketeers and mosquitos. The Moroccan invasion of the Sudan and the Songhay resistance, 1591-1612.” Journal of African History 22 (1981)

E.A. McDougall. “Camel caravans of the Saharan salt trade: Traders and transporters in the nineteenth century.” In C. Coquery-Vidrovitch and P.E. Lovejoy [eds.] The Workers of African Trade. (1985)