Arising of Contact, Feeling, Perception, Thinking
1) What is the name of this chain? Is it distinct from the links of
  dependent origination?


It is called 60 piyarūpaṃ-sātarūpaṃ in saccapabba of mahāsatipaṭṭhānasutta. 

It is a present part of dependent origination, nothing difference. 


  2) How can this chain and its parts best be explained?



Contact = metting of a sense + a sense-organ + a sense-data.
Feeling = result of contact.
Perception = memory of feeling for recollecting next time to
take/avoid the memorized feeling.
Thinking.



  3) Is there any book which can explain this chain, possibly in detail?


Abhidhammatthasaṅgaha Chapter 8.