1. He smiled brightly, and he extended his hand.(simultaneous situation)

2. He smiled brightly, and he extended his hand at the same time.(simultaneous situation)

3. He smiled brightly, and then he extended his hand. (not simultaneous situation)

4. After he smiled brightly, he extended his hand.
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You said 'at the same time' is redundent. If we want to say that after he smiled, he extended his hand, do we have to use 'and then'? Do you have some other expressions? Does #3 mean #4? Does #1 mean #2 all the time?)

1 is fine. 2 is wordy ("at the same time" is not needed). 3 and 4 are fine.

For 2, you could write this: He smiled brightly as he extended his hand. (The word "as" indicates simultaneous actions.)