Could you check whether QuickTime is displayed on the Add-ons page and, if so, whether it can be enabled? You can open the Add-ons page using either:

• Cmd+Shift+a
• "3-bar" menu button (or Tools menu) > Add-ons

In the left column, click Plugins. On the right side, look for Quicktime, especially toward the bottom of the list where disabled plugins may be found. If it's there, can you change the permission from Never Activate to Ask to Activate?

Note that the System Detail next to your question list doesn't show the QuickTime plugin as installed and enabled.

First off, thanks for the responses, and my apologies for being fairly slow at all this!

When I go to Plugins, Quicktime doesn't appear anywhere on the list. (Which seems odd, since it's the machine is less than a week old, and everything I'm reading says it comes as part of it.) And when I check "Get Add-ons," it's not listed there either. I know it's installed on my machine, since it functions in both iTunes and Safari. (And appears in "applications.") But when I try to add it manually, it doesn't even give me the option.

Okay, this appears to be an new El Capitan thing. You need to take special action to enable the browser plugin according to an Apple support article:

A small number of older webpages still rely on the legacy QuickTime 7 web plug-in to play media. In OS X El Capitan, this plug-in is no longer enabled.

Source: If your web browser says that it's missing the QuickTime plug-in - Apple Support

The article goes on to describe where you can find the plugins and move them to a folder that Firefox will check. Does that help on your system?

That worked! A million thanks! Can't tell you how much I appreciate that. I've never used a support forum like this before... but I certainly will in the future. Thanks again!