Since you have already sent the payee an electronic copy of the 1099-NEC, there should be no need to mail a paper copy.
When the IRS accepts the 1099-NEC you submitted, you will get an email that says just that, a separate email for each 1099-NEC you submitted. No, there isn't anymore you need to do.
No idea whether you'll need to submit anything to your state. Here in Calif, the IRS will forward the info to the Calif Franchise Tax Board. But different states may have different filing requirements.
You report your business expenses on your tax return, Schedule C/E/F for individuals, 1120 for corporations, etc. irrespective of the 1099-NEC amounts. Not all the business expenses are reported on a 1099-NEC.
