Most probably Google is getting the correct information, what you are seeing regarding search queries are partly a result of Google no longer always supplying this information to you.

The issue with the backlinks can probably be attributed to the fact, if I remember correctly, that Google Webmaster Tools has both your HTTPS and HTTP as separate entries. Have you added both to your account?

The quickest way to confirm the indexing is going well is to perform searches on Google with the flag `site:mysite.com` to see what is and isn't being indexed. You can also perform these using keywords to verify certain parts of the site you are most concerned about.

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**EDIT:** We're getting to the limit of my experience so I could be mistaken here.

You might consider:

 1. Switch over to fine grained redirection to the relevant SSL page, so `http://mysite.com/specific/sslpage.html` has a 301 to `https://mysite.com/specific/sslpage.html`

 and

 `https://mysite.com/specific/nonsslpage.html` has a 301 to `http://mysite.com/specific/nonsslpage.html`

 Then you preserve the link juice for your old links.

 2. Create a sitemap with all the pages served under both the sites, in this way Google will pick up all the pages whether they are indexing http or https.

 3. Move all your open pages to SSL.