The Xscope mirror, a free tool works without needing VNC but you need a paid companion app on the Mac to send the data.
I prefer to use screens on the iPad. VNC is the way that OS X sends it's video remotely to another display. If you don't use it, someone else needs to re-write that entire stack and it works very well for local networks.
As for why the app I recommend, it has an observe mode and seems to scale the screen much better than the Xscope app since it is designed primarily for designing, not mirroring. The same program works on iPad, iPod, iPhone of all sizes and retina or not displays. It is very fast and has an observe only mode so that you don't accidentally send touch events.
If you do decide to use it on the go for remote access, there is a free location app (similar to how Back to My Mac works) and works over ssh tunnels if you prefer more security than speed. It has great soft keyboard support and of course works with Linux and Windows as well. There is nothing I can say bad about it in terms of missing features os bugs or design.
It's a quality piece of software all around.
If you really must avoid VNC, there is another option. The Xscope app for software designers has a free companion app that runs on the iPad and will mirror your screen. It's more designed for counting pixels and looking at mockups you design on the Mac for eventual use on the iPad, but you can use it as a mirroring solution. It lacks any input from the iPad and simply sends the screen from the Mac to iOS.
It also is very high quality software - and is well designed, all around and fast.
I know push works for iCloud email accounts (@me.com), but as for the other ones, I'm not positive.
You can configure your email notifications by tapping Settings > Notifications > Mail. After you are there, you will notice a list of email accounts. It appears that you have to customize each individual mail account separately. Tap on a email account, and you can configure it to alert you exactly the way you desire to be notified.
You can also use VIPs (under the list of email accounts mentioned above, or accessed by by tapping the arrow to the right of the VIP mailbox just under All Inboxes in the Mail app) to customize notifications depending on if particular senders are consistently urgent.
Best Answer
At the moment the way iOS is set up is that if you want to send or share something it will only go through the Mail app. It's a bit of a bummer, but that's how things are at the moment. There are slight work arounds, so if you try to attach within an app it uses that app, but in all other cases by default it will always use Mail.
In regards to your Gmail not working, try referring to this article to add your Gmail account to mail. There are some things that have changed recently with Gmail on iOS devices. It's not longer setup through exchange so it might not work as effectively as you would if you used the Gmail App. Your mail will not be pushed automatically you will have manually open the Mail app and refresh to receive new mail. Another bummer.
I hope this information helps!