Yes, as long as the URL requested is within the same domain and path defined in the cookie (and all of the other restrictions — secure, httponly, not expired, etc) hold, then the cookie will be sent for every request.
Yes, as long as the URL requested is within the same domain and path defined in the cookie (and all of the other restrictions — secure, httponly, not expired, etc) hold, then the cookie will be sent for every request.