On a module Issue page, how do you tell what version the patch has been applied in? Much of the time, take this field collection migration issue for example, when I find an issue with a drupal module, there has already been an issue filed, and in some cases the issue has a status of Closed (fixed).  In the sidebar there is a version mentioned.  It seems likely that that version is the one you set when you file the issue, specifying which branch of the code you saw the issue in.

When the issue has been Closed (fixed), does that mean that if you grab the specified version, that the code will include the patch?  Is there any way of knowing when a fix has reached the stable branch?
As mentionned by Eyal a patch is always applied to the -dev version of the module. That's already a good stuff to know. 
So yes, if a patch is committed you can download the -dev version specified in the issue (or apply the patch separately if needed). 

Usually the patches included in a new release are listed in the changelog:
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note:
normally once a patch is committed the issue status becomes "closed (fixed)" and you will ind this kind of details in the latest issue comments: 


In the sidebar of the presentation page of the module the "View pending patches" can be helpful to check if the issue you are looking for is still opened or fixed. There is also the "Views commits" link for a quick overview. this page provides a detailed list of the patches implemented (issue date, version , description, id). 

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The official doc https://www.drupal.org/issue-queue/status could also help you to understand the process.