Plugin-container.exe is part of Firefox, not Outlook Express. If you are trying to fetch an embedded video in a message, that might use the plugin container if Firefox is your default browser, but for links and images I would not expect that to be used.

The website may be using a plugin like Silverlight or Flash Player to download specific data.

What kind of data would they download? Would they download it away fropm my PC or download it onto my PC?

The data could be an image, but more likely video or sound. Downloading means that something is going to your PC. What do you mean download away from the PC?

"download away from the PC" would be uploading a file to a web server to attach a file to an email or forum post or web page.

Some websites use a plugin like Flash or Silverlight for uploading (multiple) attachments.

What's that "specific data"? The contents of my emails? Yet another reason to stop using Windows or even quit using computers.

Outlook Express communicates with the internet and renders HTML message content using IE components, not Firefox components. If you exit Firefox before checking email, and restart to a blank page as follows:

Start > Run > firefox.exe "about:blank"

Does plugin-container still show activity?