It does index them, but that would depend on the type of page the connection is on. It won't index logins that use https.
It does index them, but a lot of sites tag https content no follow for obvious reasons (it can be an invitation to hacking attempts).
I have one site redirecting from http to https by default. The first months Google indexed all pages and showed in SERP with https. Then, the representation of pages gets to "www.site.com". All pages are still indexed with high SERP ranking.
check Paypal site, HTTPS and indexed I don't think HTTPS will make your site better, however, it can indicate you have a SSL and a unique IP
obviously the whole point of a HTTPS site is that it is a secure network, which means that if the URLis restricted then it won't get indexed in google. However even HTTPS sites normally have a standard HTTP entry page which then navigates to a secure HTTPS connection.
https is only a secured method for making a private connection between visitor and website. So Google should index those website too. https is only more secure than http websites.