What was life like in Korbach, Hesse, Germany during the 1700s?

"Protestant area" is always a very relative term in these central areas of Germany, especially in Hessen. Territorial ownership changed often during the centuries, and especially during the thirty years war and the years leading up to it, reformation and counter-reformation followed each other in short order. As a result, the next catholic parish was never far away. And if there wasn't a dedicated catholic church, practitioners often struck a deal with protestants to use their church for the occasional church service (Simultaneum).

In the case of Korbach in protestant Waldeck, it is near the border to catholic Westfalia. The next larger catholic city was Brilon (see below for road connection), but even nearer was Medebach, 15km to the southwest. The German WP article on Korbach also notes that several villages on or near the road to Medebach, and which are now incorporated with Korbach, were "almost completely" (Hillershausen) or "predominantly" (Eppe, Nieder-Schleidern) catholic. They all are within walking distance of two to three hours, so it would have been possible to get there for Sunday service if you were dedicated to your religion. If not, you would have been visiting for christmas, easter, pentecoast and a summerly parish fair (Kirchweih), at least.

The route to Cologne is very much dictated by geography. The Sauerland mountains and the Biedenkopf to the west and southwest impede traveling, especially in the rough winters. The only logical choice would have been to cross the Upland range near Willingen, 580m above sea, on the way to Brilon. I am pretty sure the road would have been more or less the same as today's still mountainous B 251, minus the modern bridges.

From there, you could follow the Ruhr river, passing along the narrow valley between Meschede and Wickede, then along the wide lower river floor to Schwerte. It is easy to then cross into the Wupper valley (the street to Barmen (Wuppertal) is even called "Kölner Straße"). Maybe someone else can comment on road conditions. As a general rule, gravel roads only started to be construced during the course of the 18th century for major connections.

Edit: I finally found a contemporary (british) map showing road connections, probably postal routes. It shows a street that leaves the valley floor at Schwelm (east of Wuppertal) in a steep rise and passes through Lennep, along the B 51 through Wermelskirchen to Mülheim. From Wermelskirchen, it was called "Wermelskirchener Chaussee". The construction of the modern gravel road was finished in 1775.

Edit 2: It turns out the is a direct Hanse route from Korbach to Cologne. It is part of the medieval trade route Leipzig – Kassel – Köln. Through the Sauerland, it is called Heidenstraße. It has two major drawbacks, though: It passes very much over the top of the mountains, reaching 750m near Winterberg, and much of it it was a hollow way, completely unfit for anything but men and donkeys.

Crossing the river would have been possible in Cologne itself with a huge reaction ferry, the Deutzer Gierponte.

The choices would have been traveling by foot, hitching a ride on an oxen cart, or a stagecoach. Riding a horse would not have been an option in the mountain parts, for heavy luggage you would have used a donkey. Whether there would have been a coach for the first leg to Brilon, I very much doubt. Imagine an unmade road, rutted down by heavy carts, and navigating inclines of up to 10%. Walking or hitching a ride would have taken more or less the same time on the 35km track. My estimate would be eight to twelve hours, depending on the weather. Spending the night up on the range would not have been an option, Willingen and the neighbouring villages had been mostly destroyed during the seven years' war.

If the map really shows postal routes, from Arnsberg there would have been the choice to take a coach to Cologne via Werl and Unna, but I share your doubt a day labourer would be inclined to finance that. On the other hand, as part of the cost of marriage, who knows?

On the second last leg, along the Rhine river, there would have been the additional choice to get a ride on a transport barge. But since it is upriver, this would have taken just as long as walking the 20 km. The last 6 km from Cologne to Efferen, apart from navigating the city and its toll stations, is an easy walk of an hour and a half.

My estimate for the time taken would be around six or seven days, with up to ten hours spent walking per day. Possible overnight stops would have been for example Brilon (35 km), Meschede (20 km), Neheim (25 km), Schwerte (30 km), Schwelm (30 km) and Mülheim (40 km). Taking the Heidenstraße, stops could be Winterberg (30 km), Schmallenberg (20 km), Elspe (19 km), Meinerzhagen (30 km), Wipperfürth (20 km) and Mülheim (38 km). Taking a coach from Arnsberg to Cologne could shorten it to four days.