So once a site has HTTPS the credentials are safe from there?
Yes and No. If it steps down to HTTP and pass the cookies in HTTP, its still vulnerable to session hijacking. For instance, you might think that static images does not require HTTPS, but the request to static images will still contain the cookie header and if it is transmitted in HTTP, then it is vulnerable to session hijacking.
Yes, there are sites that goes between HTTP and HTTPS.
When you're going through HTTPS are the cookies being sent through a the SSL tunnel? Is this right?
Yes.