Are there any other rural tramways in the world, like the Coastal Tramway in Flanders?

The Valley of the Thur Tram-Train in Mulhouse, France, is a 22 km line, mostly on single tracks after leaving the city.

Cab ride

Florian Fèvre, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

There are a lot of these in Switzerland.

BLT Line 10 is 25 km long, and has some very rural sections, with single track running and even a short part in France, making it the longest international tram line.

There is Bernmobil line 6, otherwise know as the "Blaue Bähnli" (Little blue train)

In Switzerland the distinction between Tramway and Railway isn’t always hard. Examples are the Forch Railway from Zürich to Forch and on to Esslingen. There is the Waldenburgbahn, which recently was converted to meter gauge.

Other lines of interest are the AVA lines from Aarau, with some interesting wrong side of the road street running…

Other lines:

Solothurn – Niederbipp – Langenthal

Limmattalbahn

Salt Lake City has 3 TRAX lines. They are more suburban, but the POV videos show that some parts are totally rural. https://youtu.be/ZncXSpAc4SU https://youtu.be/aY8YgW63So4

Perhaps not quite what you mean, but the SGB, Stoomtrein Goes-Borsele, a museum track, is legally a tramway (in Dutch). Its track is 14 km long.

Another tram that runs more than 10 km through rural areas is the Berlin tram line 88 which connects rural suburbs or rather villages southeast of Berlin to the suburban rail station Friedrichshagen, which is already at the outskirts of Berlin. Here is a picture by flickr user Sludge G:

Long stretches of the line follow highways crossing forests and fields between villages; here is a youtube video. I know it’s not quite the Flanders case of a tram connecting several towns; instead, it connects villages with a city’s periphery. But it is a tram, and it crosses fields :-).

Chemin de Fer de la Baie de Somme has some sections running on the street, and is partly a coastal route.

https://www.chemindefer-baiedesomme.fr/fr/chemin-de-fer-de-la-baie-de-somme

Another heritage tram system in the UK is the Seaton tramway. It’s quite short at three miles, running between Seaton, Colyford and Colyton in East Devon’s Axe Valley, travelling alongside the River Axe estuary.

https://www.tram.co.uk/

There’s also a 13km long line between Liberec and Jablonec nad Nisou in Czech Republic, though part of it is currently out of order due to changing the rails from narrow gauge to standard gauge.

Another example from Germany is the Kirnitzschtalbahn near Dresden.

I am not sure exactly how rural it has to be to qualify as "rural" nor exactly how you define a "tramway".

Several potential examples from Germany and Austria:

• The Woltersdorf tram line (Wikipedia article, OpenStreetMap) starts at a railway station in eastern Berlin and continues into a suburban village just outside of Berlin. If you want to visit that, then you should be relatively fast, because the current (almost-vintage) rolling stock will be replaced with more modern rolling stock starting later this year according to this article.
• Right next to it, the Schöneiche bei Berlin tram line (Wikipedia article, OpenStreetMap) is very similar, its rolling stock is already now a lot less interesting than that of the Woltersdorf tram line though.
• In Innsbruck, tram line 6 (Wikipedia article in German, OpenStreetMap) is a part of the Innsbruck tram network and runs mostly through a mountainous forest, not really along roads.
• Also in Innsbruck, the Stubaitalbahn (Wikipedia article, OpenStreetMap) is a former independent railway that is now operated as part of the Innsbruck tram network. It too mostly doesn’t run next to roads, but does use tram equipment.
• Another user already mentioned the Gmunden Traunseetram (OpenStreetMap) which was created by connecting the Gmunden tram line (Wikipedia article) with the independent railway Traunseebahn (Wikipedia article). I am relatively confident, however, that on the rural part that used to be the Traunseebahn, it is not legally a tram, but a railway that uses signals like a railway does. The rolling stock is one of trams.
• If you liked the previous example, then you might even like the nearby Attergaubahn (Wikipedia article in German, OpenStreetMap). This is not really a tram at all, but it uses the same kind of rolling stock as the Traunseetram.
• Last, but not least, the Badner Bahn (Wikipedia article, OpenStreetMap) connects the Vienna tram network with a short section of urban tram in Baden. On the part between Schedifkaplatz stop in Vienna and Leesdorf stop in Baden, it is legally a railway that uses railway signalling (even some freight trains occasionally run on it, or at least used to), but it does mostly run alongside a major road. In the town of Guntramsdorf, it runs on a road used by cars despite being a railway and not a tram there. Note, however, that it mostly runs through relatively ugly and boring industrial and suburban areas.

Not quite a tram, but the world’s longest trolley bus line runs an astounding 86 km (53 miles) from Simferopol to Yalta in Crimea, through rural and mountainous terrain. Simferopol is in the middle of Crimea, and Yalta is on the mountainous southeast coast (hence the trolley-bus, as a train or tram was impracticable). Yalta is the location of a major allied World War II conference.

The trolley bus is quite beloved by locals and is even deemed a tourist attraction – though difficult to visit since 2014, due to the situation. Russia would only permit travel into Crimea from Russia, and Ukraine considers doing so without permission to be illegal entry, affecting future visa rights into Ukraine.

While possible military action may effect its north end at the republic’s capitol of Simferopol, that city has a large trolley bus network that will surely be repaired. The meat of the route is in the mountains along the southeast coast, of little military interest.

Source: A. Yavin photo April 2014. An interurban trolleybus on the way from Simferopol to Yalta (Bogdan T601 of Krymtroleybus), between the Angarsky Pass and the town of Alushta. Crimean Peninsula.

Line 6 of the Straßenbahn in Darmstadt, Germany leaves the city in the south and runs on old railway tracks on to the town of Alsbach. It follows the Bergstraße for around 10 km through the countryside, with the Odenwald mountains to the east, and the wide Upper Rhine plain to the west.

(Original image description: Am 29.4.2016 ist ST 14-Tw 0782 mit SB 9-Bw 9447 zwischen Jugenheim und Alsbach unterwegs.) The mountaintop in the distance is the Langenberg, in front of it you can make out the Lufthansa congress center at Seeheim, well known as the original venue of the Seeheimer Kreis.

The Stadtbahn system around the German city of Karlsruhe might qualify:

While the local transportation consortium KVV also includes an actual tram system in the city of Karlsruhe, Stadtbahn trains (whose line numbers start with S) in the KVV area, though serving as medium-range transport, share tracks with the tram while inside the city center, and share tracks with regular trains when travelling to other nearby towns around Karlsruhe.

Public Domain image from Wikimedia Commons

At least according to the Wikipedia article, the total network track length is at least 500 km. On its way, these trains connect larger cities, but also travel through smaller towns and the countryside.

I could also find a YouTube video, on-ride style, of a 1h 15min portion line S4 (depicted above), which may give an impression if this train is sufficiently "rural" while it’s not inside a city:

https://www.youtube.com/watch?v=9XyC2kRasn8

Digging a bit further, it seems like the concept for the type of tram you are looking for is actually called tram-train, and the transportation network in Karlsruhe is described as having been among the pioneers in implementing such a system.

I might take another picture when I’m there the next time. For the time being, this one on WP, taken in the city of Heilbronn, of a tram-like train with destination Karlsruhe (a different city, about 60km away – here’s another on-ride video of that track), should also be sufficiently illustrative.

There’s the tram line Thüringerwaldbahn (German Wikipedia) in Germany with a length of about 21 kilometers that travels from Gotha to Bad Tabarz in the Thuringian Forest.

The line is famous with train spotters as various trains are used.

The Tatra Electric Railway in Slovakia also fits your description. It has some scenic views and stops in small towns near the Tatra mountain range. These stops serve as start points for hiking in the area.

Since you clarified, I think I have one or two somewhat weird contenders for your list.

The Molli in Bad Doberan–Heiligendamm–Kühlungsborn

This narrow-gauge railway is about 15 km long. It is run with steam trains that leave from a dedicated platform at Bad Doberan mainline station and for a lot of its path runs through rural countryside towards the seaside towns of Heiligendamm (known for the G8 summit in 2007 and its posh seaside hotel) and Kühlungsborn. On the Kühlungsborn end, it does run along and over a street or two for part of its journey.

The main reason for adding it to your list, however, is the part through Bad Doberan, where it connects through the town centre, through pedestrianised streets and streets with car traffic just like a regular tram. There are two stops in town. At one, you board the train at street level, iirc, the other one has curbside access level with the bottom steps of the heritage cars.

(Clicking on images opens a larger version)

The Molli is best accessed from Rostock central station (Hbf) in the North-East of Germany (Mecklenburg-Vorpommern), taking a local train to Bad Doberan station and boarding right there. It was included in the €9 ticket last summer so I am assuming it will be included in the €49 ticket subscription that you can buy this summer.

The Nordhausen tram which continues to Ilfeld Neandertalklinik

This is more of a classic tram, especially since it connects to the proper tram network of (small town) Nordhausen in Thuringia. One line of this tramway though continues out of town northbound into the Harz hills (or mountains but to me they are too low to qualify as mountains). Regular service extends to Ilfeld Neandertalklinik on a single-track non-electified metre gauge railway line.

At Nordhausen station, trams from the town to Ilfeld arrive using electric power, turn on their diesel engine, retract their pantograph and diesel out North. On the return journey, the opposite happens.

(Pictures link to larger versions)

If you are so inclined, the route to Ilfeld directly connects to the remainder of the Harz Narrow Gauge railway system (Harzer Schmalspurbahnen). They run steam-powered trains and sometimes diesel-powered railcars to Wernigerode, the top of the Brocken hill (1141 m above sea level) and Quedlinburg.

Their railcars also connect to Nordhausen tram at the station square stop, but I am not sure if they also continue into town. Railcar pictured below at Eisfelder Talmühle station.

This railway and tramway was also included in last summer’s €9 ticket so I assume it will be included in the upcoming €49 ticket. Nordhausen station is a connecting hub with lines to Erfurt, Göttingen or Kassel (both ICE connection) and Halle (Saale).

Sweden’s Inlandsbana

The 1300km Inlandsbana that runs up through the middle of Sweden might meet your criteria well enough. While it’s a train rather than a tram, with stations and platforms, it’s very much a trundle-along-and-watch-the-world-around-you type train rather than a get-from-A-to-B-asap type train. It stops at various towns and you can book package journeys that include hotels, or just buy a season ticket and hop on and off as it suits you.

The Operator’s website is available in English.

Image source

Another route that might fit your criteria is the Traunseetram, an 18.6 km line connecting the towns of Gmunden and Vorchdorf in Austria.

The equipment used would probably qualify as a tram rather than a train (although I’m by no means an expert), its tracks run on the street for the section in Gmunden, and along a road for the last few kilometres into Vorchdorf.
It also offers a nice view of the mountains and the entire route most certainly qualifies as rural:

Image source

For the most part, however, it does not run along a road like you specified.

You can find more information on the operator’s website (German only), the (very short and uninformative) English Wikipedia entry, or its more extensive German version.

Blackpool Tramway
The question is very broad, but another coastal tramway is at Blackpool, UK.
At 18 km with 38 stops, it isn’t as long as the Coastal Tramway in Flanders.
Wikipedia says

Blackpool Tramway runs from Blackpool to Fleetwood on the Fylde Coast in Lancashire, England. The line dates back to 1885 and is one of the oldest electric tramways in the world. It is operated by Blackpool Transport Services (BTS) and runs for 18 km (11 miles).

By Steven’s Transport Photos – https://www.flickr.com/photos/33216596@N02/51309706473/
CC BY 2.0, https://commons.wikimedia.org/w/index.php?curid=108024089

Not as long as the "Coastal Tram", but the Borkumer Kleinbahn on the Borkum island might be of interest to you:

The Borkumer Kleinbahn is a 900 mm (2 ft 11+7⁄16 in) narrow gauge railway on the German island of Borkum in the North Sea. It is the oldest island railway (German: Inselbahn) in Germany, beginning operation in 1888.

The German wiki page has more information on it. Apparently during summer they they run this tram/train creature on it once a week:

_{Von Peter Hudec, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=15468329}