In Word you've got the option to 'Paste as plain text' (or 'Paste without formatting', not sure of the exact name in English). That way, you can easily paste text from any website without markup.
I don't think it should be the responsibility of the website to implement a 'copy without markup'.
Other trick: paste text with markup in Notepad first. Then select it in Notepad and copy again. The clipboard now contains only plain text.
Of course it's hard to educate the users, but unless there is a very good reason why people would be copying texts from the website, I wouldn't make a custom implementation for it. Those implementations, especially when you override default behaviour (capture Ctrl+C), will probably not work well or at least the same in every browser, and they are just a fix for your website, not everyone else's. Also, such a feature might annoy other users who do know how to handle text with mark-up. So I think it's better to let the users figure out themselves.
As for hijacking the clipboard, if you copy HTML, the browser actually copies the content in multiple formats, like HTML, RTF and plain text. If you had the means, you could simply remove the HTML and RTF version from the clipboard and let the plain text version remain. But I'm pretty sure you don't have that much control over the clipboard from the browser.