Re: no alert sound
You say you often don't get an alert which suggests it's not settings related as if you get it sometimes it must be set correctly.
I have the same issue only not just for mail but for every alert sound including the camera shutter sound. Basically it doesn't always play an alert sound for the likes of email, messages, camera shutter, lock sound....come to think of it this is mainly iOS stock apps that fail to audio alert every time. The camera shutter sound nearly always doesn't play the first time it takes a photo and I nearly always take it again because I am paranoid it didn't take until I then check the camera roll and both images are there. I've seen it first hand where my iphone 6 was on the coffee table unlocked and on the home screen and it lights up and the visual banner notification appears for a new iMessage but no sound! I then send one to myself to test it and low and behold the sound alert then plays!
Others have reported this also, hopefully for you this is the issue you're having too?