See MADAMIRA :

http://camel.abudhabi.nyu.edu/madamira/

Yes but there is also Diab's tool and others. Besides, Stanford POS tagger does not perform lemmatization automatically, unless we create one's own code.

You can use MADA+TOKAN, the tool of Nizar Habash et al. It's a freely available tool. Take a look to this document.

http://www.elda.org/medar-conference/pdf/24.pdf

Yes I think that MADA is the only operational tool for full morphological disambiguation, but the problem is that it was trained on a modern corpus. We need a tool for classical Arabic.

Also Eric Atwell developed several tools with Kais Dukes within the quran corpus.

See MADAMIRA :

http://camel.abudhabi.nyu.edu/madamira/