Learn Italian: Stavo per guardare il riflesso di entrambi i gemelli. - I was about to look at the reflection of both twins.

In Italian, when entrambi (both) is followed directly by a noun, you must include the definite article for that noun. So "both twins" becomes entrambi i gemelli (literally "both the twins").

If I wanted to use tutti e due instead of entrambi, how would the sentence change?

It would mean exactly the same thing, and the grammar stays the same: tutti e due i gemelli. Just like with entrambi, you must keep the definite article i right before the noun.

Does the di before entrambi translate to "both of"?

No, the di here simply connects "the reflection" to the twins: "the reflection of". In English, we sometimes say "both of the twins", but in Italian, you never add a preposition after entrambi to say "both of". You just go straight to the article and noun: entrambi i gemelli.

Why is stavo used here instead of a completed past tense like the passato prossimo?

The structure stare per followed by an infinitive verb means "to be about to do" something. Because being "about to" do something describes an ongoing state or a past intention rather than a finished, completed event, stare takes the imperfect tense (stavo).