I have to say, I had a few sales yesterday when I should have heard the notification on the app, because my phone was right by me, AND I was on my laptop when the sales came in, and sometimes I got the cha-ching, and other times I didn't. And I had just deleted and re-loaded the app this week, due to lack of notifications.
* Not sure about the "not opening for sold items" - you can always try to delete and re-install.
I have an iPhone and when this happens to me it’s usually because I accidentally turned the phone sound off. I notice when I see the strip of red on top left side button. Red shows up when ringer is off.