In the UK and some other Commonwealth and Anglophone countries you might hear people talking about the 'corn exchange' this refers to the old practice of farmers brining grain in to the cities and towns for merchants to purchase. Generally a 'corn exchange' will be situated inside a large building, rather than being outside, but they are made up of individual market traders operating from stalls like your typical market square.
"Markt" and "Marktplatz" have different meanings and both terms are commonly used in German.
- "Markt" is a place or an institution where people can sell and buy things of a common category. This may be food (farmer's market = "Wochenmarkt") or shares (stock exchange = "Aktienmarkt") or property (real estate market = "Immobilienmarkt") and so on.
- "Marktplatz" mostly denotes a special location in a town or village where the "Markt" takes place e.g. every Saturday (or has taken place in former days). The "Marktplatz" has its name also on the remaining six days of the week. The "Marktplatz" in Karlsruhe, Germany, is the center of the city: http://ka.stadtwiki.net/Marktplatz The "Marktplatz" will keep its name even if there is no "Markt" at this square any longer.
May I puzzle you? The term "Marktplatz" is also used for a virtual place in the web where a specific market takes place: https://de.wikipedia.org/wiki/Marktplatz (see the second meaning). An example of such a "Marktplatz" is http://quoka.de/ Look at its footer: "Dein Marktplatz für gebrauchte & neue Artikel."
Yes, exactly. "Der Markt findet auf dem Marktplatz statt." will be well understood in German. "Marktplatz" is the location and "Markt" is the weekly recurring activity at this location.
You can say: "Ich gehe jetzt zum Markt auf dem Marktplatz." or "Ich gehe jetzt zum Markt, der auf dem Marktplatz stattfindet." This sentence is very clear. But in spoken German it is also common to say: "Ich gehe jetzt noch schnell auf den Markt und hole Tomaten." Here the activity "Markt" borrows its name to the spot where this activity takes place at the moment. So you can say "Ich gehe auf den Markt" instead of "Ich gehe zum Markt". Sorry, German is hard ... :-)
You mention that the place may be named Marktplatz even if there is no longer a market. In English you would often refer to the open area in the town centre as the square or town square. Is Marktplaz the equivalent in German or is there another term you might use for an open area in the middle of town?
"Festplatz", "Rathausplatz", "Kirchplatz", "Dorfplatz", "Messplatz", "Richtplatz" are frequently used in German towns and villages as names of such places. It depends on the concrete situation and (former) usage of that place which of these names is used. I don't think that there is a generic term similar to town square. "Stadtplatz" is not a German word.
I think that "Where is the market place?" is a valid translation of the sentence "Wo ist der Marktplatz?" even if www.deepl.com marks "market place" as less common than "marketplace" for US English. So I think it would be a good idea if you report your solution to Duolingo next time.
"Marktplatz" can also be used as a generic term. Yes, there are market squares which are officially named "Marktplatz", but that doesn't mean that there aren't ones with a different official name, which can't still be referred to using the term "Marktplatz".
Duden simply defines "Marktplatz" as: "[zentraler] Platz in einer Stadt, auf dem Markt abgehalten wird oder früher wurde".
The Marktplatz in Karlsruhe (where I come from) in the southwest of Germany is famous for his pyramid. This is both, a monument and a tomb, where the founder of Karlsruhe is buried. http://de.wikipedia.org/wiki/Marktplatz_%28Karlsruhe%29 The Marktplatz is also used as a market place every morning as you can see in the second picture of the linked Wikipedia article.