Having trouble with the mail apps

I am experiencing a number of issues using both the iOS mail app and the Gmail app.

My Gmail settings are like this: I have 2 gmail accounts, account1 is general; account2 is for school, job search and whatnot. While on the desktop, I only check my account1 mailbox and have set it to check and fetch mails from account2. I have set account2 as the default "send mail as" while sending mails from the account1 mailbox. The settings carried over and worked flawlessly on my Android Gmail app.

Now I just made the switch over to iPhone and have been experiencing a number of issues with each of the following apps:

Apple mail app:

1. No push notification (despite of it being turned on). Screen won't come on when a new mail is received.

2. Cannot send mail as my default account2, unless I add another email account, but that would result in me receiving 2 copies of each email that's sent to account2.

3. Whenever I send a message, I automatically get a cc in my inbox.

iOS Gmail app:

1. Whenever I send a message, I automatically get a cc in my inbox.

I think I prefer the Gmail app over the Apple one due to the lesser number of issues, but seeing that I am on an iOS device now, I do wish to make an effort to use the default mail app as it feels more cohesive to the rest of the iOS vibe.

I don't know if anyone can help me in resolving those issues, at least the one plaguing the Gmail app.