Learn Dutch: De Rijn is een lange rivier. - The Rhine is a long river.

In Dutch, names of rivers, seas, and oceans always take the definite article de. Therefore, it is always de Rijn, de Maas, or de Noordzee. While most countries and cities don't use articles, bodies of water are a strict exception.

Is Rijn treated as singular or plural for verb agreement?

It is singular, which is why it uses the singular verb is. Individual rivers are always singular in Dutch. You only use plural verbs for geographical proper nouns that are inherently plural, such as mountain ranges (de Alpen zijn) or island groups.

Do we capitalize the article de mid-sentence when referring to the Rhine?

No, only the proper noun itself is capitalized. If this wasn't at the beginning of the sentence, it would be written as de Rijn. This differs from some city names where the article is a fixed, capitalized part of the name, like Den Haag.

As a quick refresher, why does lang become lange here?