Learn Swedish: Jag hör vargen yla när månen går upp. - I hear the wolf howl when the moon rises.

Because hör (hear) is a verb of perception. When you hear, see, or feel someone doing something, Swedish uses the perception verb + the object (vargen) + the bare infinitive (yla without att). It works exactly like English: 'I hear the wolf howl'.

Would it be wrong to say Jag hör vargen som ylar?

That is grammatically correct, but it means something slightly different. Jag hör vargen som ylar means 'I hear the wolf that is howling' (focusing on identifying which wolf you hear). Jag hör vargen yla means 'I hear the wolf howl' (focusing on the action you are perceiving).

Why does the moon 'go up' (går upp) instead of using a specific single verb for rising?

Swedish commonly uses the phrasal verb gå upp (literally 'go up') for celestial bodies rising, like the sun and the moon. You will also see this same phrasal verb used for people waking up and getting out of bed in the morning!

What happens to the word order if we add inte (not) to the second part of the sentence?

Because när månen går upp is a subordinate clause, Swedish uses the BIFF rule where sentence adverbs come before the verb. If you wanted to say 'when the moon does not rise', inte would step in right before the verb: när månen inte går upp.