Learn Hardcore Icelandic: Við vonum að veðrið verði gott.

Because this is a subordinate clause after a verb of hoping. Icelandic normally uses the subjunctive mood (viðtengingarháttur) in such clauses. Verði is the present subjunctive of verða (“to become/be”), and it conveys a desired or uncertain outcome. Using the indicative verður would sound more like a factual assertion (“that the weather will be good”), which clashes with the idea of hope. The most idiomatic choice after vona is the subjunctive: að veðrið verði gott.

Is að here the same að as the English “to,” or does it mean “that”? Does it force the subjunctive?

Here að is a complementizer meaning “that,” introducing a content clause. It is not the infinitive marker “to.” The conjunction að by itself does not force the subjunctive; it’s the meaning of the main verb (vona “to hope”) that selects the subjunctive in the subordinate clause.

Why is it veðrið (definite “the weather”)? Could I say just veður?

Icelandic typically uses the definite form when referring to “the weather” in a given time/place, much like English says “the weather.” So veðrið is normal. You will see the bare noun veður in set phrases and existential statements like Verður gott veður á morgun? (“Will there be good weather tomorrow?”), or in compounds (e.g., veðurspá “weather forecast”). In this sentence, where a specific stretch of weather is in mind, veðrið is the idiomatic choice.