When I create a new document, this option is available, but how to disable it, when the document is already created?
I can select shapes and disable it via the transform panel but not for newly created shapes.
When selecting symbol this option is greyed out, Illustrator realigns the paths anyway.
In the top right corner fly out menu of the transform panel, you can select or deselect "Align New Objects To Pixel Grid" globally for all future objects/paths (see below image)
For each already created path/object use the checkbox in the transform panel itself
The paths within the symbol might be set to "Align To Pixel Grid". Disabling it for the specific paths might be ignored anyway, so it's necessary to dispatch the shape from the symbol, disable the align option and recreate the symbol. Illustrator bugginess FTW.
you're probably best to do this when setting up a new document in the document set up dialogue box. i think you can also turn it off in the preferences which would be a good idea because in my experience it does nothing but cause trouble!