No, you can set cookies to HTTP only, so Javascript can't access them. JS injected into a domain can do something with your permissions for that domain by making more requests to the target domain that have the cookie attached, but at the moment that's basically how the Web security model works, so that is in some sense not a hole. But the injected JS can't steal the cookie and send the contents somewhere else.
Why would an XSS attacker need to steal a temporary session ID from a cookie (which will probably expire soon), if they can just highjack the session right there on the spot from the user's own browser?
Not exactly 'anytime' because the session will expire as soon as the user logs out. Even if the user doesn't log out, the session will typically timeout on its own anyway (at least if the auth is implemented correctly).
