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The story centers around a group of people that get lost during in the woods and -for no apparent reason- suddenly find themselves in 13th century Europe (the Netherlands, to be precise). They don't know how they got there and there doesn't seem to be much they can do about it, so they stay in a small town for at least year till they figure out what has happened to them. In the meantime they try and make a living for themselves. Some of them find work, some use their skills and knowledge from the future to make a living.
One of them is a woman who has a dark skin.

My questions are:

• How would people in the middle ages react to a dark-skinned person? There is no harbor in town and it's quite small, so chances are they've never seen someone with a dark skin. However, they might know of their existence because of the crusades: the story is placed at the start of the 13th century in the middle of the 5th crusade (not that the protagonists were involved, but a lot of dutch people participated in it so it must have been a thing at the time).




• What would be a good occupation for my character? The original idea was to have her take on some medicinal activities, maybe even in disguise, but this was also at the time of witch burnings and such and I'm not sure how well people would've reacted to that.

Any recommendations for research literature are also welcome!

She can be an Ethiopian Christian Pilgrim.

https://en.wikipedia.org/wiki/Council_of_Florence

https://en.wikipedia.org/wiki/Christianity_in_Ethiopia

In 1441 some Ethiopian monks travelled from Jerusalem to attend the
Council in Florence which discussed possible union between the Roman
Catholic and Greek Orthodox churches. The arrival of
the Christian monks caused something of a sensation. It began two
centuries of contact in which there were hopes to bring the Ethiopians
into the Catholic fold...

Would a small town in the Netherlands be aware that there might be a distant foreign Christian kingdom? Yes they would!

https://en.wikipedia.org/wiki/Prester_John

Prester John (Latin: Presbyter Johannes) was a legendary Christian
patriarch, presbyter (elder) and king who was popular in European
chronicles and tradition from the 12th through the 17th centuries. He
was said to rule over a Nestorian (Church of the East) Christian
nation lost amid the Muslims and pagans of the Orient, in which the
Patriarch of the Saint Thomas Christians resided.

Your traveler can say she is on a pilgrimage to visit some Christian site and get a blessing to take back to her ill husband. The people will be interested in her because of her exotic provenance but more comfortable because she is christian and also interested in the local holy site. She can make her living telling stories of distant lands and peoples. Probably no-one will pay her gold for stories, but they will put her up and feed her.

The link to the Moors on Wikipedia is already provided in comments so I won't put it in here again, but it's important to note 2 things that are germane to your question;

1) The first 'Re-conquest' of the Moors in Spain occurred in the 13th Century

2) It was the sacking of the Moors that led to many of the developments in mathematics and science within the Christian worlds from that time through a couple of centuries because the Moors had been spending considerably more effort in enlightening themselves than the Christians of the time.

As an educated guess, your dark skinned time refugee would struggle socially, but the people of the Netherlands would express little concern or surprise at the technological or mathematical expertise of your traveler, assuming that she was a Moor. This would actually be a benefit to her because it's a convenient excuse as to why her language, customs, and occasional gaffes (like using Okay instead of All Right) may be forgiven.

To be blunt, in that era she's going to get more resistance from being a woman than she is being black but if she knows what she's doing she could easily set herself up in midwifery. That wouldn't be seen as that big a leap for women at the time, she'd already know more than the women of the day in many respects, and perhaps most importantly, if she saves mother and child from some awkward situations a couple of times over, all thoughts of witchcraft will probably go away on the grounds that there's no deodorant quite like success in this field.

Personally, I'd keep her away from technical or engineering fields - don't let her be a blacksmith or design buildings or do accounts for the local businesses. Their math was very different to modern mathematics for a start, and standing out in such fields, especially as a woman, is an ideal way to attract said unwanted attention of witch hunters.

Be a leper.

Good excuse to cover up. Nobody will touch you and in fact will give you stuff to leave. If the woman has the assistance of the group, she should be able to survive without too much hassle

Another other option is a pilgrim. The group can speak for her and with extensive robes remain hidden.

Final option is to be blatant and open by being a travelling merchant from a far away land.

If it's only the woman who is dark skinned, the easiest solution is to pretend she's a war booty. One of her male colleagues will pretend he captured her in war in the south. If her actual boyfriend is part of the group that's even better. An independent woman if middle-ages was more suspicious than a dark skinned person. Most women had three choices - marriage, monastery or prostitution. Whether your character is actually married to her "captor" or just a concubine doesn't really matter.
As a bonus, being a captive from faraway lands she can explain some of her knowledge. People can be bit more forgiving of her exotic wisdom, as long as she's under control of her white man. Medicine is a good idea, but don't push it too far. Being a Moorish witch is not much better than local witch.