>>For a while I thought there's something to be set in about:config (just type that as an URL and see what you can set there), which would prevent this (which is actually a security measure - nothing coming from the web touches your clipboard), but can't find it.
>
>Lots of different settings there, but do you know if I can set it up to always open with the tabs I opened in my last session? That was a feature in Opera, which was very convenient.

Download Firefox 2.0. It has sessions buil-in. I still prefer to use Tab Mix Plus extension. See http://sergey.berezniker.com/x/ZgE