1. He pretended he was carrying the glass necklace /and started down the stairs.

Did the two actions happen at the same time(before 'and' and after 'and)?

2. Pretending he was carrying the glass necklace, he started down the stairs.

3. While he pretended he was carrying the glass necklace, he started down the stairs.

(Are both sentences the same as #1?)

1. Yes, since both verbs are in the simple past tense, both occurred at the same time.

2 and 3 - yes, they mean the same as #1.