Concerning the logo, this mainly depends on the template and technology used to specify a logo.

There are different method depending on the case.

Case 1: The logo is defined by a configuration parameter in the template and is located into a subdirectory of the /templates directory.
In this case, you can either create multiple logo file name depending on the website into the /templates/xxx/ directory
OR make the template directory specific for each site.

Case 2: The logo is stored in the /images directory and you are using the Media Manager to select the logo file name.
In this case, it is possible to modify the default path used by the Media Manager in going into the Media Manager and modifying its configuration to use for example your "images_specific" directory.
So that, you will continue sharing the /images and use specific one with the other /images_specific directory.

Case 3: The logo is managed via a module (ie. custom HTML)
In this case, you have to take care about the editor that is used to edit the images.
ie. JCE may automatically modify the path in the <img src > tag by removing the http (or https) value. They cleanup the path that sometime is not whished.

Case 4: If the logo is managed via a CSS value for which the name and the path can not be modified.
In this case, you have to modify the CSS to make it more flexible.

Case 5: The logo is a hardcoded path and filename somewhere in the template.
In this case, it is possible to modify the path using the free Search/Replace plugin.


In your specific case, I suspect that you are in the case 2.