I changed the shortcut to close a window to Ctrl+W and to close all windows to Ctrl+Shift+W. Close all windows works fine, but Ctrl+W selects the word currently under the cursor but does not close the window.
When I right-click a tab it say Ctrl+W is the shortcut to close it (And also Ctrl+S to save and Ctrl+Shift+W to close all). Why is only the close window shortcut not working?
In Visual Studio (VS 2015 in my case but it's similar down to VS 2010 at least) keyboard shortcuts may have a different meaning depending on the context in which they are executed.
Click Tools / Options / Environment / Keyboard to look up or define shortcuts (you already did that probably). What I called "context" is selected in the combobox labelled "Use new shortcut in:". Most likely you defined the shortcut in "Global" but you want it to work in "Text Editor". In the latter Ctrl+W selects the current word.
Redefine Ctrl+W for "Text Editor" and you should be fine.