What Layout engine do the common Web Browsers use?

IE

IE  uses Trident layout engine.
http://en.wikipedia.org/wiki/Trident_(layout_engine)

FireFox 

FireFox uses the GECKO layout engine
http://www-archive.mozilla.org/newlayout/doc/layout-2006-12-14/master.xhtml
https://wiki.mozilla.org/Gecko:Home_Page

Opera

Opera uses the Presto Engine
http://www.opera.com/docs/specs/presto29/

Safari

Safari uses the Webkit Layout Engine
http://www.webkit.org/

Chrome

Chrome uses the Webkit Layout Engine
http://www.google.com/chrome/intl/en/webmasters-faq.html

What are the Javascript Engines (browsers) out there? and Who is using what?

IE

IE and Edge uses the Chakra(JScript) engine. 
"JScript is the Microsoft implementation of the ECMA 262 language specification (ECMAScript Edition 3)".
http://msdn.microsoft.com/en-us/library/hbxc2t98

Chrome

"Chrome uses the V8 Engine. V8 is Google's open source JavaScript engine".
http://code.google.com/p/v8/

FireFox

Firefox uses SpiderMonkey (Tracemonkey). "SpiderMonkey is Mozilla's JavaScript engine written in C/C++. It is used in various Mozilla products, including Firefox, and is available under MPL/GPL/LGPL tri-license." 
https://developer.mozilla.org/en/SpiderMonkey

Safari

"The Nitro JavaScript engine is an advanced bytecode JavaScript engine that makes web browsing even faster, and it’s what powers Safari on the Mac, PCs, and iOS devices."
http://www.apple.com/safari/features.html

Opera

Opera uses an Javascript engine called carakan (futhark old version).
http://my.opera.com/chooseopera/blog/2009/02/05/meet-carakan-and-vega