Learn Dutch: De vlucht gaat door, tenzij de storm hevig is. - The flight is going ahead, unless the storm is severe.

This is because of tenzij (unless). It is a subordinating conjunction, which means it introduces a dependent clause. In Dutch dependent clauses, the working verb is pushed to the very end. That is why it is tenzij de storm hevig is instead of tenzij de storm is hevig.

Does gaat door come from a specific verb, or is it just the word door added to gaan?

It comes from the separable verb doorgaan, which means 'to go ahead', 'to continue', or 'to take place'. Because De vlucht gaat door is a main clause, the verb splits: the main part (gaat) takes the normal second position, and the prefix (door) goes to the end of that clause.

Why doesn't the adjective hevig get an -e at the end here?

Adjectives generally only get an -e ending when they are placed directly before the noun they describe (like de hevige storm). In this sentence, hevig comes after the linking verb is. When used later in the sentence like this, the adjective always stays in its basic, dictionary form.

Could I say behalve als instead of tenzij?