When playing Forza5 sometimes the picture on the screen jumps from side to side. Here is a ...

Looks like you've got Kinect head tracking on. Go to settings and turn off head tracking and that should stop it. If that's already off, would seem you have a dodgy right analogue stick on your controller.

Nothing happens. The invites come in on the right side of the screen. When I click ACCEPT, my cursor jumps down to the bottom of the screen but the game doesn't start up or, if it's already booted up, it doesn't join the game session in play.

You cannot snap games to the smaller side (right side) of the screen from my understanding. You should be able to play CoD on the bigger side (left side) and watch TV on the smaller side though.

My day one controller works fine but the casing on the let hand side is a bit lose and you can see the difference when you look at it from the back the right hand side panels are more together than the left hand side ones. Can hear it sometimes when I am
playing games.

When you say side by side do you mean on the same console split screen or on two different consoles/tvs?