Learn Swedish: Hunden rusade väl ut ur grottan? - The dog rushed out of the cave, didn't it?

In Swedish, compound prepositions are often used to show both the direction of movement and the relationship to the space. ut indicates the direction "out", and ur means "out of" (originating from inside something). Together, ut ur specifies moving from the inside of an enclosed space (the cave) to the outside.

Could I say ut från grottan instead of ut ur grottan?

You could, but ut ur is much better here. ut ur specifically highlights that the dog was inside the three-dimensional, enclosed space of the cave before coming out. ut från (från meaning "from") is more general and just means moving away from a starting point, without necessarily emphasizing being inside it first.

Where does the "didn't it?" come from in the Swedish sentence?

That comes from the little word väl. It doesn't literally translate to "didn't it", but it is used when you assume something is true and expect the listener to agree. In English, we usually express this feeling by adding a tag question like "didn't it?" or "right?" at the end of the sentence, or by using a word like "surely".

Why is väl placed right after the verb rusade?