I would like to know, is there a way to control your iPad with your Mac remotely (be able to see and manipulate iPad's desktop on your Mac)? I know that with the TeamViewer or other programs you can control your Mac with your iPad but not vice versa.
It is not possible to remotely control the screen of an iPad unless you jailbreak and install a VNC server as MattDMo suggested. This is because the security model in iOS does not allow an app to reach outside of its sandbox to control other apps or the springboard home screen. Apple does not provide any remote control functionality at the iOS level either. There is no way to do what you are asking for without jailbreaking.
Mobile Device Management (MDM) solutions do not allow remote control of the screen on iOS either. MDM can push configuration profiles to control settings on an iPad, or prompt the user to install an app from the app store, but they can't control the screen. Airwatch offers a screen control option, but this only works on platforms that support it (not iOS). I work with several MDM vendors, including Airwatch, and this is not possible because Apple does not make this functionality available in iOS.
There's a tool (made for testing purposes) with the ability to do that, first released on July 18, 2014: iOS Gateway
There are a couple of options. iKeyboardRemote is an app that claims to do just what you want, and is free. However, after installing it, it seems like it's limited to pressing the home button, looking at Pictures, and playing music.
You can remotely control your iOS device's screen using one of many Mobile Device Management solutions, such as AirWatch for example. However, these are enterprise-level tools that are typically used in the workplace. There is no easy way to do this as a home user, unless you enroll in such a program.
You can also jailbreak your iPad and install a VNC server.