A scary scenario no doubt. This is why I always keep my webcam/mics unconnected unless in use, I'm always cautious of what software I run and etc. At the end of the day though I'm not a likely target for such things, I am after all a mere male with above average computer skills. I worry more for my family because of their children. I try to "educate" them on the dangers of using P2P software, running their computers without anti-virus/anti-malware software, etc., etc., but I can't hold their hand every time they turn on their computer.
To answer your question, Linux is most definitely not invulnerable to this kind of thing. "Software" (aka RATs) exist for virtually every OS out there (even your cell phone). There's no need to be paranoid though. The fact is these guys want easy targets, so they prefer casual computer users who in turn prefer Windows. So it's unlikely that you'll be targeted since Linux represents such a small number of desktop computers and in general Linux users are also more knowledgeable and thus more cautious about executing foreign binaries/scripts and etc.
This could change in the future. As more people - in particular casual users - use Linux it becomes more and more of an attractive target for these kinds of exploits.
Stay away from bad sites. Keep your browser (and any other software) up to date. Don't run scripts/binaries from untrustworthy sources. Be cautious of all foreign material (emails, websites, etc.). Even a seemingly
benign JPEG attachment could be dangerous. Keep these things in mind and you should be reasonably safe.