'what with ~' 'to bat someone's eyelids' meaning


I'm reading Harry Potter and the Half-blood Prince and run into this sentence.

...he(Harry) was soon surrounded by a mob of people congratulating him. What with trying to shake off the Creevy brothers, who wanted a blow-by-blow match analysis, and the large group of girls that encircled him, laughing at his least amusing comments and batting their eyelids, it was some time before he could try and find Ron.

I've got some questions:
1. What does 'What with' mean? How can I understand this phrase?
2. Does 'least amusing' mean 'not amusing'??
3. What does 'batting their eyelids' mean? As far as I know, 'not bat an eyelid' means 'to show no surprise or embarrassment'.
4. What does 'it was some time before~' sentence mean? Can you rephrase this sentence?

Thank you in advance.
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