MacBook – Youtube in Safari cause system crash

The log files that you've posted aren't providing much good information regarding the crashes. The best clue from the logs may be this line (which could be indicative of OpenGL graphics context problems):

9/10/13 10:42:57.764 AM SystemUIServer[271]: It does not make sense to draw an image when [NSGraphicsContext currentContext] is nil.  This is a programming error. Break on void _NSWarnForDrawingImageWithNoCurrentContext() to debug.  This will be logged only once.  This may break in the future.

However, your description of the problem symptoms (the screen going black/grey; a red line appearing during boot) may point directly at graphics (video card) issues. Although it may be possible for you to perform some tests to pinpoint the problems, it may be worth having the machine examined by a qualified technician.

Using the Apple Hardware Test may help you to diagnose why the crashes are occuring. Alternately, there are some good freeware diagnostics available for OS X (Rember will test the RAM in your Mac).

If the problems are graphics/video related, you may be able to reproduce the issues by pushing the machine's graphics processor to perform. This can be accomplished by using some sort of high-end graphics application (one that heavily utilizes OpenGL). A potentially helpful option may be Cinebench

It's also possible that you're having problems related to graphics context switching. I might recommend trying to disable that option in the Energy Saver System Preference to see if the symptoms subside. More info on How to set graphics performance.

It is a case of desecrete GPU failure. Most of the Early and Late 2011 MBP are prone to this failure. The culprit is the AMD GPU. See more at the links below.

http://www.digitaltrends.com/computing/suddenly-bunch-early-2011-macbook-pro-owners-running-video-problems/

http://www.cultofmac.com/262861/early-2011-macbook-pros-dropping-like-flies-heat-issues-blame/