The handsome man smiled at her, then winked.
This is good usage. The comma represents the missing 'and' in your next example:
The handsome man smiled at her and then winked.
Both are correct, just different variants.
I'm not keen on 'The handsome man smiled at her, and then winked' as it seems more a stylistic choice than a grammatical one. It brings back the old 'use a comma for a pause'.
She walks inside then pushes the door closed.
Some will say this is okay: two predicates to one subject, so therefore no 'and' used. Some will argue it's not, that you need the coordinator between 'then':
She walks inside and then pushes the door closed.
As this is just a style issue, consistency would be the answer (pick one method and stick to it)
She walks inside and pushes the door closed.
I think the arguement would be can that you can't do both actions together (walk inside AND push the door). You'd need the 'then':
She walks inside and then pushes the door closed.
She takes a deep breath, looks back, then descends into the darkness.
Just looking at it grammatically:
Punctuation suggests you're doing three actions in a sequence here.
1: takes a deep breath
2: looks back
3: descends into darkness
She takes a deep breath, looks back and descends into the darkness.
Punctuation suggests two actions:
1: takes a deep breath
2: looks back AND descends into darkness.
Well, if 'descends into the darkness' is a state of mind then, okay if it's she's moving down into darkness, I'd question 'how' if she's looking back. Lol.
Either is correct depending on when you want the 'action sequence' to happen. But again, if it's a state of mind that she's descending into, you can have her look back and descend into darkness. If it's two actions: looking back, descending, I'd question whether you could do both at the same time without falling over.