There is not much to say about this, but no, the reference comes from a second-hand account of german city ordinance from a sixteenth century. There is no overarching work about swimming and bathing prohibition (to my knowledge) between sixteenth and nineteenth century, but it did happen locally, notably for two broad reasons:

(1) Dangers of drowning,
(2) Nudity, public morality, etc.
And yes, swimming lessons were a thing, and people argued that teaching is more effective than prohibiting in terms of (1), at some places more successfully than others.

I know Ljubljana (nowadays Slovenia) had such prohibitions of swimming/bathing in the eighteenth century for the latter reasons (2), so these highly depend on city/local governance, but they typically predate nineteenth century. One would have to look through local records or local studies for specific jurisdiction though, so I cannot give an estimate, and probably no study to that effect exists. Also note that some of these restrictions might still exist to this day in some form in some countries on local levels for the same reasons, like swimming in Paris Sienna is prohibited (although these are also frequently associated with pollution safety, tides safety, sea and river traffic in later centuries up to this day) . I hope this is not too lackluster, but I do not know whether more could be effectively said at this point.

Yes and no. It seems that the series strongly exaggerate various facets of immigration to the point that the portrayal of the fresh immigrants verges on that of 'stupid foreign yokels'. Even if one could possibly accept that a lot of new immigrants to the USA, usually coming from the rural European background did not know how to ford rivers, what is a specialist knowledge even today, it is hard to imagine that people coming from poor areas, where hunger in winter was an ongoing concern, were unable to ration food or make sure to take care of their equipment, especially given that they have already successfully taken a long journey that included crossing the Atlantic. Thus, I would advice caution.

Now, people were very rarely learning to swim, because this skill was rarely useful for most people. As majority of the society consisted of agricultural workers and craftsmen who were usually living in a single area throughout most of their life, they had very little occasion to swim, unless they lived in the vicinity of larger river or a lake. If memory serves, the character saying the story about flogging the bodies comes from a mountainous region of Germany, where rivers are relatively small, cold and have a fast rapids, making them relatively unpleasant places to take a swim. Washing clothes or the body was usually done in a shallow parts of the unregulated river and streams, and thus it did not require people to swim.

It is true that in 19th century, in some places, authorities made it a misdemeanor to swim in the rivers. In German context, a good example is the regulation introduced in Stargard in Pommern (now Stargard, Poland) in the first half of the 19th century that forbade swimming in the river Ihna under an extremely severe penalty of... a reprimand. This ruling was most likely caused by the fact that Ihna was a major transportation route and random swimmers posed a source disruption and were endangering themselves, as the river traffic in that period was relatively high, especially near the cities, due to convenience of the water transport (if you look at the pictures from 19th-century cities located on the rivers, you'll see that the rivers were positively crowded with ships and boats). As even an accidental collision with a boat could have easily proven fatal, and the natural bodies water (especially rivers) can be dangerous to even a capable swimmer, the rulings were introduced to discourage people from swimming in such places. This is pretty much the same type of regulations that mandate wearing a helmet when riding a motorbike or forbid crossing the street in a place other than a pedestrian crossing (or crossing it on a red light). The traffic comparisons are not without merit, as drowning was not a very unusual kind of death in 19th century, as in England it claimed the lives of 2500-3000 people annual, a number similar to the number of people who perish in the traffic accidents in largest European countries today.

The first modern handbook of swimming has been written in 1538 by Nikolaus Wynmann, a Swiss–German professor and published in 1538 under the title Colymbetes, sive de arte natandi dialogus et festivus et iucundus lectu (The swimmer, or a dialogue on the art of swimming and a joyful and fun lecture). It was followed by De arte natandi (on the art of swimming), that also happens to be the first illustrated instructions book written in English, published in 1587 by Everard Digby. Unlike Wynmann, who focued on practical side of swimming, Digby described also the physics principles of swimming. Of course, both books were aimed by the relatively small group of people who could not only read and take instructions in writing (what was not that uncommon), but first and foremost, could have afforded it. But the swimming got more and more foothold in England, with Dr. Robert Wittie of Scarborough publishing materials where he advocated bathing in seawater for health purposes in 1667. In early 18th century some English schools, most notably Eton and Harrow recommended swimming lessons in the 1720s, with the first official training pool (then a mere natural pond) being designated only in 1780s. So it can be said that swimming, although by no means popular, slowly gained recognition a good two centuries before the period series in question are set in.

Swimming was considered a good form of physical training for young people also in Germany, where in 1793 Johann Christoph GutsMuths [sic!] published his Gymnastik für die Jugend (Gymnastics for the Youth), a sports manual that also included a chapter on swimming. Muths then followed the swimming theme in his Kleines Lehrbuch der Schwimmkunst (Small Textbook on the Art of Swimming) that has been published in 1798. It is worth noting that GutsMuths must have been aware that by advocating swimming in the rivers that were, after all, public places, he needed to take into account the modesty standards, so he also described preferred swimming clothing resembling boxer shorts, incidentally creating the first recorded modern swimming garment. This strongly suggests that swimming in rivers and lakes was not something that contradicted social mores of the era, provided the modicum of modesty was observed. New fashion of swimming in the seaside spa centres, initiated in England, very were quickly adopted in German states, with the first swimming and bathing center being established in 1793 in Doberan, a seaside city in the Duchy of Mecklemburg (now Bad Doberan, a district of Rostock). The notion that Germany was a country where swimming was banned in 1883 or shortly before is the more absurd given that the German Swimming Federation, an official body training swimmers for local and international competitions has been created in 1882, possibly in a response to a similar organization being formed in London two years earlier.

There is a claim that the swimming was banned and bodies of the drowned were whipped in accordance with the local regulations introduced in Ingolstadt in 1530. This can be possibly linked to the aforementioned book Colymbetes, where Nikolaus Wynmann, a student and later professor of antique languages at the University of Ingolstadt noticed that in his native Switzerland kids often swim, swimming is much less common in Germany and recalls the situation where a 'dead body of a drowned boy was beaten with rods', although it is not clear whether this was an anectodal occurrence or a result of any sort of custom or law. Even if it was a standing law, it is hardly possible that a random German immigrant could have recalled such an obscure ruling enacted in one Bavarian city three and a half century before.

So, to sum it up, Germany was no different from other countries in the 1880s when it came to swimming, which was still not that popular among the general population, with most people having little need, incentive or an opportunity to learn this skill.

Chaline E., Strokes of Genius: A History of Swimming. Reaktion Books, London 2017.