I dont use any task killers and have very few additional apps installed and have seen several small issues with the alarm clock.
First was the occasional ringing at a minute or two late, I have it set for 7:30 am M-F and have had it go off at 7:31 or 7:32 a couple times, now the other day my wife calls me at 7:37 and it decides to ring then so both the phone and alarm are going off at same time, I dont sleep through alarms or use snooze so I know thats not the issue...
I would hope whatever this is gets resolved as even the cheapest POS phone from a decade ago had a working alarm clock you could depend on, of course I have also never heard of a phones camera focus being dependent on some inner calender clock either....
Was it from Gmail? Or was it tied to another account? If Gmail your email will be pushed to your phone as soon as the account receives it. Remember there are several factors from my keyboard to your screen that could have delayed the process. Try sending an email from one account to another on your home computer and you will see there can be time diffences. The phone is not to blame, it is a solid piece that people just havent taken the time to figure out. They want to turn on the phone and have it do everything for it. Just take a deep breath, count to ten and then read up on tips here on how to use your phone.
Try removing your app killer app.
If google thought killing apps was important they would do it automatically or provide an app