Learn Danish: En ny orkan kan opstå ude på havet. - A new hurricane can arise out on the sea.

No, that would completely change the meaning. opstå is an inseparable prefixed verb meaning to arise or to originate. If you split it into the separable phrase stå op, it means to get up or to stand up. The position of the prefix dictates the definition here.

Why do we use ude instead of ud?

ude is the stationary form of the adverb, used here because the hurricane is forming at a specific location out on the sea. You would use the directional form ud if the sentence involved movement from the land toward the ocean, such as blæse ud på havet (blow out to sea).

Why is the preposition på used with havet instead of i?

Danish uses på havet to refer to the sea as a general geographical surface or area, which is where weather events happen. You only use i havet when describing something physically submerged in the water, like a fish or a sinking ship.

Why doesn't opstå have the infinitive marker at before it?

opstå is governed by the modal verb kan. In Danish, just as in English, modal verbs always take the bare infinitive of the following verb, meaning you drop the at.