Sorry to hear. But maybe a simple reason.
First, AFAIK clicking in the backend on either Invoice or Shipping Invoice never sends them, only shows them.
That some mails are being sent, shows that the trouble is pretty surely caused by setup/config, the more as us others don't have the same trouble as you.
The one thing that should definitely trigger mails is an order being created, and the change of order status in the backend - the latter at least if you do select "notify".
If this doesn't happen, you should
verify that email settings are made and correct:
mainly in -> System -> Configuration -> Emails (all okay for your needs, and are there settings that your ISP might not like, leading to him blocking your outgoing mails?),
but also in -> System -> Emails (all wanted mails published?).
What you tell one can say as much as I did. Better guesses could possibly be made if you showed us screenshots of your settings. And again, make sure that your host (server) is totally okay with both your mail settings and content.
Hope it helps. Otherwise come back. Best with pics.