Logo behavior in responsive wordpress theme

This website [login to view URL] is based on wordpress with this theme [login to view URL] I would like the Logo (the man over main menu) is only displayed on the homepage. For all other pages i would like that instead of the Logo there is a blank space of the same heigh. I would like that when in home page i click on the "X" that compress the side bar, Logo fades out and disappears. And when I extend the side bar clicking on "+" the Logo fades in and becomes visible. This is a responsive theme so I'd like that in the mobile version (when interface structure changes) the logo never be displayed, neither in homepage nor in other pages. At last when these tasks are finished i need to transfer the web site in an other domain.