Hello I lost my invoice for the MacBook Pro purchased in Apple Store Fifth Avenue NYC. Can I ask for a re-sent on my e-mail?

Yes. Ofcourse. However you might have also received the same in your email.

You usually get an email invoice. If you can't find it. They can resend it to you via email. Always take email version even if you want a printed. You can get both

I don't know,but so long as your phone number and/or credit card info is in their system they can always look it up as proof your purchase was made at the store.On your next purchase ask them to send your invoic/receipt via email.