you've a lot of videos on this site with russian teachers (our profiles), so it's quite easy to watch them and decide if we speak the way you have mentioned or not.

Absolutely not. I have never heard a russian person speak English as those american actors do.

 For some reason movie makers overview the fact that foreiners might have a slight accent.

 Even speaking about thick accents, they make very noticable mistakes. 

1. Mouth too stiff. Russian speakers do move their jaw and lips less than English speakers. Yet we still do use them. Actors trying to fake russian accent look almost like amateur ventriloquists. Hence, all vowels become very bleak, even [i] and [e], when in Russian they tend to sound really soft, weather you want it or not. 

2. Obviously you change your voice as you change a language. But those actors usually put on a very weird rasping/guttural voice, which is not a characteristic feature for russian speech. 

3. Usually, Russian speakers don't roll their [r]'s when speaking English, but try to pronounce them American way, just really badly. Also I have never heard a Russian speaker who could pronounce [th], but could not pronounce soft [r]'s and had to roll them.

In most cases you can tell, weather the actor is faking Russian accent or not. Only on two occations I was fooled into thinking that there is really a Russian actor playing a Russian guy.

I have no idea where this fake Russian accent came from. It's in all the films and TV shows, and is instantly recognisable as a "Russian" accent. Yet, I have never in my life heard a Russian speak with accent even slighly resembling it. The real question is how did it get invented and why is it perpetuated?!

Almost every non-native speaker whom I hear sounds like shit when speaking English. Same with those who are learning Russian, it's natural. I've heard some Finns that were sounding much worse than we do.

I think yes! Many Russian people have a really freaky accent and they speak like robots =)