Learn Dutch: Zij springt het hoogst over de horde. - She jumps the highest over the hurdle.

Because it is an adverb modifying the verb springt (it describes how she jumps). In Dutch, adverbial superlatives always use the fixed word het followed by the superlative form of the adjective or adverb. It does not change to match a noun.

Would it be wrong to say het hoogste with an extra -e at the end?

No, it wouldn't be wrong. In everyday spoken Dutch, you will often hear an extra -e added to adverbial superlatives, so het hoogste is perfectly fine here. However, het hoogst is the strict, traditional form you are learning in this lesson.

Why use the stressed pronoun Zij instead of the unstressed Ze?

Superlatives naturally compare people or things. Using the stressed Zij emphasizes the person: she is the one who jumps the highest, compared to the other competitors. If you just wanted to state a casual fact without emphasizing her specifically, you could use Ze.

What happens to the word order if we start the sentence with Over de horde?

If you move that phrase to the front for emphasis, the verb must stay in the second position. This causes inversion (the subject and verb swap places), so the sentence becomes: Over de horde springt zij het hoogst.