a small help needed

To set up what type of notifications you hear for each app you have to do that in settings under notifications. As for Gmail email messages coming in, I find that sometimes they arrive in my inbox as they are sent but most times they don't load into my inbox until I've launched the mail app. I believe that if you used the gmail app instead of the iOS mail app your Gmails would come in as they are sent.

I have an @me account and an @icloud account and mail to those come in night away but mail to my gmail accounts usually do not.

at Khushi, go into settings, passwords and accounts. Once in passwords and accounts, go to the bottom. You should see fetch new data, go in there. Turn on push, if its turned off. Next, go into your mail account in that same section, its below the push setting you just turned on if it was off. Once there, you can select I believe, hourly, every 30 minutes or every 15. You'll recieve emails even if the app is closed. Hope this helps.