For new orders that are disappearing - that's very strange and should not be happening. I'd make sure that you don't have any additional extensions that are automatically processing orders, or moving them into a different order status after they come in.
I would recommend checking your admin area to view ALL orders instead of just ones that are pending.
I would also recommend that you check in with your payment processor to make sure that payment is going through properly, and that you have orders in your payment system. That will allow you to know / ensure that your orders are actually making it through.
For the transactional emails not showing up (order confirmation emails), you need to do a couple of things:
- Make sure Magento cron is configured and running properly. If Magento cron isn't running, the system will not send out confirmation emails.
- With Godaddy hosting, you can't just send emails via SendMail and have them go through - you will need to use their custom outbound email settings. You'll want to set your system to use the relay-hosting.secureserver.net system. More details are here: http://www.satollo.net/godaddy-using-smtp-external-server-on-shared-hosting
Also, make sure that the email address you are using is the same domain as your website; a yahoo or gmail address won't work.
And make sure you have an SPF record in your DNS zone file to give permission to the site to snd emails on your behalf.
I hope this helps!