1. None of your facebook friends can see your forum posts or google searches or anything like that unless one of three things happens:
- You click the "like" button, as then it'll be broadcasted on Facebook
- You get http://en.wikipedia.org/wiki/Clickjacking#Likejacking
- They are REALLY intent on tracking you down across the web (and even then only so much information will turn up)
That said, to address the title, which is quite different than what the OP is about: Everything you post on Facebook is effectively in the public domain, especially now that http://www.engadget.com/2012/08/16/federal-judge-facebook-profile-privacy/.
Basically when you post something on facebook:
-- First your privacy settings kick in. Let's say you're smart and restrict your profile to only be viewable by friends. It doesn't matter because:
-- your Facebook apps all have permission to all your content. Let's say you're smart and don't use any Facebook apps, it still doesn't matter because:
-- your friend's facebook apps have permissions to all your content.
Rule of thumb is: whatever you post on Facebook you should expect to eventually be seen by someone you'd rather it not be seen by, because more than likely it will. Best off not posting anything you'd rather not have anyone in particular see.
Good example of it is the http://www.bilconference.com/videos/privacy-is-dead-eric-gradman-cloudmirror/
Facebook doesn't keep track of your browsing history, so people won't know anything about your forum activity unless you specifically mention it, and there's an option to upload videos into youtube.. the only things I ever uploaded were some videos but it's not an automatic thing.
Actually, if you're logged into Facebook, http://lifehacker.com/5843969/facebook-is-tracking-your-every-move-on-the-web-heres-how-to-stop-it. If they didn't, the Facebook "Like" button wouldn't work when you wanted to click it.