Any received letters will show as "Completed" even if your app has been not been submitted and you have not paid. Mailed letters take a long time, but if you did electronic, you will get an email stating that your reference has submitted on your behalf. If they haven't received anything from that ref, then it will still show as "new". If it is electronic and they have opened it but not completed it, it will show as "in progress".
No, you don't have to wait. If you are done with your application submit it now. When your reference letters come in, they will update it and it will show as "Completed". Don't wait for your letters come in because your app still needs to be verified. 🙂 Once they receive the letters, then they submit everything to your school(s).
Good luck!