Learn Italian: Temo di scoprire qualcosa di terribile. - I fear discovering something terrible.

Because the subject of both verbs is the same person (I fear that I will discover). When the subject of the main verb and the second verb are identical, Italian replaces che + subjunctive with di + the infinitive. If the subjects were different, you would need the subjunctive, for example: Temo che tu scopra (I fear that you will discover).

Could I still say Temo che io scopra if I wanted to?

Grammatically, it is heavily frowned upon. When the subjects match, using di + infinitive is considered the standard rule. Saying che io scopra here would sound very unnatural and clunky to an Italian speaker.

Why is there a second di before terribile?

In Italian, when you want to attach an adjective to an indefinite pronoun like qualcosa (something) or niente (nothing), you must link them using the preposition di. That is why 'something terrible' translates to qualcosa di terribile.

The English translation says 'discovering'. Why isn't the Italian gerund scoprendo used?

English frequently uses the '-ing' form as a noun object (I fear discovering). Italian doesn't do this; it uses the infinitive (scoprire) as the noun form of a verb. The Italian gerund (scoprendo) is generally saved for actions currently in progress, like 'while discovering' or 'I am discovering'.