Internal HDDs will never be external. They are known as
secondary HDDs.
If it's an IDE, you will have to set it to slave mode before
putting in to computer B. Computer B should have no problems
detecting it.
The reverse, however isn't true. Older versions of Windows can't
detect newer version of Windows files. A new version of Windows has
to be installed before trying out with an older one.
You can retrieve the data, but the OS won't work because it's
corrupted. However, chances are, your files aren't, so you can
recover them after slaving it.