My notifications are not working right on my iphone??

So there are 2 problems about notifications on my iphone

1. the notification badges wont show up until i open the apps and close it. I told this to the apple guy but he said it was a bug. But my family and friends doesn't have this problem. I have all the apps ON on my notification center. I researched this but it didnt help me alot.

2. Whenever my phone is asleep, the notification shows in the lock screen. But when I open my phone, the app that has a notification opens directly to it. Its really annoying. When I researched this, i found out that this problem is rare.

My iphone is not jailbroken. I really need help, if you have any solutions please say anything you know.