Well, this would be a case of "there's the rule, and then there's all the exceptions."
An object by default would only be in existence during a single execution. (Read: Until you get to the final line of index.php, in this example). Garbage collection then cleans everything up.
An object instantiated into the $SESSION global array would (provided index.php invoked sessionstart() ) exist for the duration of the session.
An object passed as a cookie (bad idea, but possible) would persist for the lifetime of the cookie. (Actually it wouldnt - it'd get passed back and forth repeatedly, with the object's server-local existence being destroyed by garbage cleanup as usual)
Including is treated (for the most part) as though you took the contents of the included file, and inserted them at that point. Variables are in-scope, functions, etc. (The reverse is also true, if the included page defines variables or functions, they are available in the global scope of the file that did the including.)