Learn Italian: Il secchio è caduto, di modo che l'insegnante si è spaventato. - The bucket fell, so that the teacher got scared.

This is the key difference between result and purpose! Here, di modo che introduces a result—a factual event that actually happened (the teacher got scared). When this phrase means "with the result that," it takes the indicative. If the sentence expressed an intention or purpose ("so that the teacher would get scared"), it would require the subjunctive.

Could I use in modo che in this sentence instead?

Yes, absolutely. Di modo che and in modo che are generally interchangeable. However, you will often see di modo che (or just cosicché) used slightly more often for results, while in modo che is heavily used for purpose. Both are completely correct here to mean "so that".

Why does cadere (to fall) take essere as its helper verb (è caduto)?

Cadere is an intransitive verb (it cannot take a direct object) that describes a change of state or location. Verbs of movement and state changes usually require essere as their auxiliary verb in compound tenses like the passato prossimo.

Why does the past participle end in -o (si è spaventato) when insegnante ends in -e?