Qt seems to fit your requirements
Has mono/.NET bindings called QtSharp (There are older ones referenced about on the web but this is the latest AFAIK)
Has a docking example http://doc.qt.digia.com/4.6/mainwindows-dockwidgets.html you should be able to determine what is going on as the api is the same in c#
OpenGL/DX defiantly can do both. (Have a search on ...
wxWidgets has .net bindings called wx.Net which are cross platform for MS.Net, Linux & Mac (Using Mono).
Native look & feel
Free (Libra & FLOSS)
Not sure about OpenGL & DirectX contexts in the current version.
I personally have not used wx.Net but have used wxWidgets, (C++), and wxPython for years with a great deal of success....