First, I'm assuming that TeamViewer has a setting to allow it to run without popups. For the paid-for version I would consider it exceptionally unlikely they would not allow this as an option. So look through your TeamViewer settings for this, or check out the answer to that problem here: How to hide the notifications of the Teamviewer (silent use)
The basics of this solution are to open TeamViewer Options > Advanced > And enable "Automatically minimize local TeamViewer panel.
Second, run VLC as the desktop background, which is also doable in Ubuntu: https://askubuntu.com/questions/565154/how-to-set-a-mp4-as-a-wallpaper
The gist of this solution is to launch VLC from the terminal using the command
cvlc --video-wallpaper /path/to/your/video
Third, hide the Ubuntu launcher and menu bar like this: https://askubuntu.com/questions/9865/how-can-i-configure-unitys-launcher-auto-hide-behavior
Install the CompizConfig Settings Manager, under Desktop find the Ubuntu Unity Plugin and set Hide Launcher to AutoHide.
Running the video as background will make it more resilient in that interruptions on the desktop will not interrupt it or exit full screen. Then, of course, you have to hide the various UI elements so that they aren't overlaying the video.
Finally, programs popping up on login is just bad behavior in general and should never happen by default.
While the first step should negate the need for the other two, I'd personally implement all three to make the entire system more uninterruptable in general.