Go to court to file form immediately. But make sure you

have "Certificate of mailing" from USPS for your Pay and Quit letter. If not, you may have to repeat the Pay and Quit letter again and wait for 5 days.

I don't think you can personally deliver the Pay and Quit. You need a friend (not relative) or third party to deliver and sign the form of the service.

Otherwise, if you have done all above, file a form immediately and schedule court date at the same time, it could take 3-4 weeks before you appear in court.

Well, once you start the eviction process, no need to contact tenant again. However,if they pay in full plus the late fee, you must withdraw the suit.

One suggestion to you: Always charge a higher security deposit than the monthly rent. This will psychologically deter tenant from nonpayment. I charge 1.5 monthly rate for security deposit.

If you end up having to evict them, security deposit may be all that you can hope for after all is done. In your case, $500 is not enough. Had you charged 1.5x Rent, you will be in better shape.