Learn Italian: Per il malore, si è dovuto fermare precipitosamente. - Because of the sudden illness, he had to stop precipitously.

Because the reflexive pronoun si (from fermarsi) is placed before the conjugated verb. When a reflexive pronoun goes at the front of a modal verb plus an infinitive, it forces the auxiliary verb to switch to essere.

If I wanted to attach si to the end of the infinitive, how would the verb phrase change?

It would become ha dovuto fermarsi. When you attach the pronoun to the end of the infinitive, the modal verb keeps its normal auxiliary, which in the case of dovere is avere.

If the subject of this sentence were female ("she had to stop"), would the verb change?

Yes, it would become si è dovuta fermare. Because the auxiliary shifted to essere, the past participle (dovuto) must now agree in gender and number with the subject.

Why is per used here to mean "because of"? Couldn't I use a causa di?

You could absolutely use a causa del malore, but per is a very common, slightly more concise way to express the cause or reason for an action in Italian. It often translates to "due to" or "out of" (like in "out of fear").