A question asked yesterday shows some ImageMagick approaches to get to step 2.
As for step 3, there is a ImageMagick-based application named G'Mic that provides some advanced multi-step filters. The gallery shows some results that come close. However, it is in C++ so you'll probably need to compile it on your system.
Also, Googling for Imagemagick Cartoonify yields some example scripts that are worth checking out.
Also, make sure you dig into the ImageMagick examples page, one of the greatest IM resources around.
That said, I doubt whether it is possible to build a IM-based solution that works as well as the cartoonifier you show. IM is hugely powerful, but may not have the necessary degree of detailed control, individual programming and vectorization functions. But the basic functionality is possible.
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