Learn Dutch: Wij veronderstellen de gezelligheid is belangrijk.

In standard written Dutch, yes. You would normally say Wij veronderstellen dat de gezelligheid belangrijk is, using a subordinate clause with the verb at the end. However, this sentence drops the word dat (that), which is very common in spoken Dutch after verbs of thought or speech. When you drop dat, the following clause reverts to standard main clause word order, putting the verb is in the second position.

Is veronderstellen common in the casual spoken Dutch that usually uses this dropped-dat grammar?

It is actually a bit formal. In everyday casual conversation, a Dutch speaker would more likely say Wij gaan ervan uit (we assume) or simply Wij denken (we think). However, it is quite normal to hear people mixing slightly formal vocabulary like veronderstellen with informal spoken grammar like dropping dat during business meetings or academic discussions.

Why is there a de before gezelligheid? In English, we just say "coziness" without "the" here.

Dutch frequently uses definite articles (de or het) before abstract nouns when talking about a concept in a general or specific context. In this sentence, they are likely talking about the specific atmosphere of a workplace, event, or group, making de gezelligheid a tangible, specific element rather than just a vague idea.

What is the literal breakdown of the long verb veronderstellen?