1) The simpliest way would probably be to add a simple custom action 
that sends Apple + W to EyeTV. Which of course assumes that the Live 
TV window is the currently focused window of EyeTV. The more complex 
way is to write an AppleScript that goes through the list of EyeTV's 
windows and identifies and closes the Live TV window. That 
AppleScript, too, would have to be added as a custom action at/via RB

2) Again, you can achieve this via a custom action, with f.ex. this 
AppleScript:

tell application "EyeTV" 
activate 
set full screen to true 
end tell

In the custom action editor, you'd want to check the checkbox for this 
action to make it global. You can then put it anywhere in the menu and 
also map it to any button of your remote in any mapping table.

Best regards, 
Felix Schwarz