Learn German: Sie weint vor Freude. - She is crying with joy.

We use vor because crying is an involuntary, physical reaction to an emotion. When a feeling causes an uncontrollable reaction (like shaking with fear or crying with joy), you use vor. If the action is a deliberate, conscious choice (like doing something out of love), you would use aus.

Why is there no article before Freude? Shouldn't it be "vor der Freude"?

When using vor or aus to show the emotional cause of an action, we usually drop the article before abstract nouns like Freude, Angst, or Liebe. This is known as using the zero article.

Could I say "mit Freude" since the English translation is "with joy"?

Not in this context. Mit Freude translates more to "gladly" or "with pleasure" (for example, volunteering to help with pleasure). To say that joy is the actual underlying cause of an involuntary physical reaction like crying, you must use vor.

Even though there is no article shown, what case does vor trigger here?

It triggers the dative case. vor is a two-way preposition, but when it is used to express a cause rather than a physical location, it always takes the dative.