There seems to be a lot of curiosity about this topic as of late. This thread will answer most of your questions. But since I've been doing some research about executioners in medieval Central Europe, I'll try to give you some more info about this specific area and period. Extremely long post ahead.

For starters, one of the first misconceptions is that the executioners were present throughout the Middle Ages. This is blatantly false. The executioner is not mentioned in medieval sources until the XIII century. In the early and high Middle Ages, the court generally established the conditions for reconciliation between the victims and the accused: the victim of the crime or his relatives received compensation ("wergeld") that corresponded to his social situation and the nature of the offense. But even if the court sentenced the defendant to death, the sentence was not carried out by the executioner. In old German law, the death penalty was initially carried out jointly by all those who tried the offender, or the execution of the sentence was entrusted to the youngest assessor, either the plaintiff or an accomplice. Often, the convict was entrusted to a bailiff, whose duties, according to the Sachsenspiegel, included maintaining order during court hearings: summoning trial participants and witnesses to court, delivering messages, confiscating property, and the execution of sentences, although it is not clear from the source's text whether he had to do it himself or only to monitor the execution.

In the later Middle Ages, the authorities became more actively involved in criminal proceedings. Imperial law, which established universal peace, could not have ensured the cessation of blood feuds, strife and other violent acts, if the public authorities had not provided an alternative to private execution in the form of corporal criminal punishment. Law enforcement took over the initiation of criminal proceedings, investigation, and arrest of suspects. No longer relying on (traditional in the early Middle Ages) formalistic evidence, such as a cleansing oath or ordeals ("God's judgment"), the judicial authorities began investigating the circumstances of the crimes and questioning the defendants in order to obtain a confession. Therefore, torture became an integral part of the criminal justice system. In the 13th century, i.e. long before the influence of the reception of Roman law began to affect (end of the 15th century), in addition to the new legal procedures, there was a proliferation of more complex corporal punishment in Germany, which became typical for criminal proceedings throughout the early modern period, displacing wergeld as a form of retribution for a crime. Although hanging and cutting off the head remained the most common forms of punishment, wheeling, burning at the stake, burial alive, and drowning became widespread. These executions could be toughened by additional torture to which the convicts were subjected to: scourging, branding, amputation, piercing with hot rods, etc. These new procedural rules were the result of the public authorities' desire to pacify society by focusing a monopoly on the legitimate use of violence in their hands. Thus, in the XIII century, in connection with the new regulation of corporal punishment and death penalty under the law of peace in the country (Landfriedengesetz), there was a constant need to carry out more and more various torture and executions, which required already known qualifications - and then there appeared professional executioners in the public service. But the monopoly on execution of death sentences was established for them only by the end of the 16th century. For the first time in German sources we find a reference to a professional executioner in the code of city law ("Stadtbuch" of the free imperial city of Augsburg, 1276). Here he appears as a municipal employee with clearly defined rights and duties. Which leads us to the next question:

Was that their full time job? Did it pay the bills?

Yes, and no. There were plenty of duties that were "attached" to the profession, but did not involve execution itself. There were also some side jobs that weren't a part of the profession, but were common among executioners for reasons that I'll explain later (Franz Schmidt, whose diary I strongly recommend, wrote that he healed far more people than he had executed, for example).

First of all, let's define the nature of the job. The laws of the city established the monopoly right of the executioner to execute death sentences and "all corporal punishment." When he took office, the executioner signed the same contract and took the same oath as all the other officials who were subordinate to the city authorities, depending on the status of the city, either to its council or to the liege; from them he received a salary, an apartment, and other allowances on an equal footing with all other city employees. His work was paid according to a fee established by the authorities: for each execution on the gallows or on the scaffold he was to receive five shillings (this is the data from Agusburg laws, but the fee was different in different cities and at different times). In addition, the executioner received everything that the convict wore below the waist - this tradition was preserved throughout the next centuries. When with age or after illness the executioner became too weak to do his job, he could resign and receive a life pension. At the same time, at first, he had to help the new "master" who replaced him with "good advice and faithful instruction", as it was customary in all other posts in the communal administration. In many cities where there was a uniform for municipal servants, it was worn by the executioner as well. But there is no mention of masks or hats with eye slits, which are often seen in historical novels and movies, in late medieval sources.

So, the executioner was a professional of execution and torture. But since, except for the extraordinary cases of mass repression, this work did not take up all his time and did not bring him any income which could've earned him a living, the executioner, apart from his main occupation, performed other functions in the urban economy.

First, he supervised the city's prostitutes. The executioner was actually the keeper of the brothel, he made sure that women behaved according to the rules established for them by the authorities, and dealt with conflicts and disputes that arose between them and citizens. The prostitutes were required to pay him two shillings every Saturday, and the executioner was not to "demand more." Prostitutes who did not have permission to live in the city or were expelled for breaking the rules, he was obliged to expel from the city. Same with, incidentally, lepers - for this he was paid five shillings each time city taxes were collected.

The executioner's function as a keeper of a brothel seemed to have been preserved throughout the 14th century and in many towns and cities of the 15th century. Thus, in the Bavarian town of Landsberg, this practice remained in place until 1404, when the executioner was fired for participating in the beating of a competitor who had no permission to practise his trade in that town. In Regensburg, the executioner's brothel was located in the immediate vicinity of his home, and in some other cities prostitutes also lived in the exec's home, such as Munich, until the Duke of Bavaria's order in 1433. In Strasbourg, the executioner supervised not only the trade of "priestesses of love," but also the gambling house, and had some income from it. In 1500 he was suspended from this duty, but as compensation he was obliged to receive a weekly surcharge from the city treasury. In Memmingen, the authorities at the beginning of the XV century hired a special person to the post of the keeper of a brothel, but he also regularly paid the executioner a certain amount. In Augsburg, as soon as the 14th century, the executioner was not the only one who controlled prostitution: the sources mentioned a woman named Rudolphina; by the end of the 15th century, the function of a municipal brothel keeper had finally been taken over by a special official. In other cities as well, gradually, starting in the mid-15th century and especially after the Reformation, when brothels in Protestant regions were closed for religious and ethical reasons, the executioners lost their position and, with it, their source of income, which was replaced by a salary supplement.

Edit: spelling and grammar