Is it possible to broadcast video with a software player on a separate monitor connected to the laptop?

If I understand correctly the question, then connected the monitor, go to monitor properties and put a check "Extend my Windows desktop onto this monitor" (for XP), then on the second monitor, you can do anything, mode to "full screen", etc. will be applicable only to the second monitor when on the ground can be worked.
\r
\rConfigure and use multiple monitors in Windows XP

And flash video has canceled FullScreen while you work on a laptop — here is the article with the solution:
\rhabrahabr.ru/blogs/browsers/107606/

If the nvidia graphics card suddenly, then their site is the nView utility that automates the spreading of Windows, or even redirect the video stream to an external monitor.