Would a medieval village govern itself for the most part? How would the villagers go about this? Would the head of the village be the one who dealt with higher officials? Were there village heads? Are there any records on this topic?

Although the 'Middle Ages' is a rather vague concept both chronologically and geographically, from the quoted inspiration from the question (Witcher 3 video game) and the proposed time constraints, i.e. 12th-14th century, I'll focus on the village life in the high-to-late Middle Ages in Western and Central Europe that is also quite prevalent setting in the fantasy works. To summarize the answer in one sentence, one can say that in the place and time mentioned above the villages were indeed self-governed.

What is important to keep in mind is that within the discussed period and place, i.e. 12th-15th century in Western Europe that largely coincides with the concept of 'high-to-late Middle Ages' large and still increasing part of villagers could have been considered a kind of sole traders. They were renting a land granted by the landowner and were generally free to do whatever and however they please, provided that they rendered required goods and services as a form of recompense or taxation. This largely meant that the external governance of the village was not that much needed, although it would be incorrect to say that it did not exist. Landowners were generally interested in the income from tributes and rent what was linked to the efficient management of the particular plots, but the latter was generally a given as it was in the best interest of the tenant farmers to manage them as efficiently as possible, because they were entitled to all the net product gathered from them i.e. everything that was produced minus the necessary taxes both monetary and delivered in nature. Thus, with the villagers themselves taking care of the village life, owner of the land was largely involved only whenever some kind of problem have arisen.

Please note, that the direct involvement of the landowner was usually limited to the lower nobility, who had small land granted by their seniors in a single region, and thus were able to supervise them personally. The situation was more complex for the lords who held significant holdings that were not only physically large, but could have also been scattered around the country or province, making direct management difficult if not downright impossible. Such affluent landowners usually resolved this matter by appointing their direct representatives in every holding, usually a knight or a priest in the secular or ecclesiastical holdings respectively. This representative, usually called a bailiff in England and hailing from the lower nobility or rich peasants was tasked with supervising the work of the serfs working lord's fields, sale of the goods produced there and procurement of all goods and services required for a manor to operate - from tools to candles to equipment of the facilities owned directly by the lord, such as windmills or breweries. Direct control over the serfs was usually delegated to another official, known in England as 'reeve'. This was almost always one of the local free tenants, most often than not one holding and working a large lot, usually with the help of additional personnel, and thus well-versed in management of work. Reeves were also responsible for the practical supervision of all elements of domain that could have required repairs (bridges, fences, walls, buildings) and sale of the goods produced in the lord's fields. It can thus be said, while bailiff was supervising the economical side of the lord's domain, reeve was doing most of the legwork. Last but not least, it was usually the bailiff or reeve (sometimes both) who physically collected all rents, taxes and goods due from the tenants. It is worth noting that these positions, especially that of a reeve, certainly gave at least some space for creative embezzlement, as Walter of Henley in his influential economical treatise Le Dite de Hosebondrie written in early 1280s notes, that it is prudent to check the actual number of grain bundles and other items in the granary after receiving the official report.

It should be stressed however, that the role of the bailiff and reeve or their equivalents was largely focused on the management of the landowners' own domain and the work of serfs, so their official interaction with free tenants was quite limited and the latter were largely left to their own designs. What is equally important is that the villagers were usually not really interacting with any sort of 'government officials' save for the actual owner of the land.

The idea of 'village elder' or any form of a more or less formally elected person governing the settlement is also not really historical construct in the context of the high-to-late Middle Ages. Not counting the landowners or their representatives, villages, especially those inhabited by free tenants largely did not have any formal organizational structure and functioned more as set of separate homesteads with the villagers knowing each other. This meant that each family was predominantly focused on working their land and making sure their obligations to the landowner were paid in full, but in any matter that included anyone else, they were simply getting along just fine through casual, everyday interaction. This was facilitated by the size of the settlements, that in the high-to-late Middle Ages could have consisted of anything between few and few dozen of homesteads and 20-250 people, including children. Religious ceremonies and other customs, such as annual procession along the borders of the village lands, practiced in some places around the Ascension Day were celebrated by all the inhabitants of one or more villages, effectively cementing the communal ties.

Any kind of quarrels were usually resolved amicably, but if such a solution was unavailable or when the problem was criminal in nature, villagers had to resort to the official way, and bring the issue before the person holding the judicial privilege in a given village. Most often than not it was a local landowner, but it could have been their senior or another superior official that reserved the judicial power (and income generated) from themselves. Depending on the agreements between the local landowner, local custom and character of the case, plaintiffs were able to bring the issue to the proper authority directly, present it to the lord's representative who would then arrange all matters or wait until the lord will visit the village to resolve the issue.

So the villages were indeed largely self-governed, with the external influence being largely limited to the representatives of the landowner who were tasked with collecting taxes and other due obligations, and sometimes dealt with external visitors, such as e.g. royal tax collectors. But for the most of the time, villagers were resolving all their daily matters themselves, usually between the families or within the community.

Lamond, E., Walter of Henley's Husbandry, Longmans, Green & Co., London 1890.