I don't use Facebook but, as far as I know, Ant Video Downloader should be capable of capturing the video from a Facebook page (because it can capture video from almost anywhere on the web). It's an extension that's only available for Firefox (which I'd recommend using) or Internet Explorer:
http://www.ant.com/video-downloader
Once it's installed you'll see a new icon at the top right of your browser window:
http://tsguide.e-dv.co.uk/uploads/2/8/6/4/2864874/5680743_orig.png
Clicking on the little downward arrow allows you to enter the 'Preferences' section, so that you can choose where videos are saved. (If you don't make your own choice they go into a default 'Ant Videos' folder inside My Documents > Downloads). Then, when viewing a video, clicking on the main button should save the video file to the selected folder.
To email the video, create a new email as usual. Click on 'Attach', 'Attachments', 'Attach file' or whatever your email system calls it. Navigate to the location of the video file, double-click on it and then send the email. (That assumes that the video file size isn't too big for your particular email service. If it is you'll need to upload it to a file-hosting site and then send a link to it by email but, for now, I'll assume that won't apply).