Is there any survey paper focusing on the study of DIFF PL LIP TOP categories?
I learnt of much of their properties when looking at the stuff on microbundles. A very clear source for lots of that is J. Lurie, Spring 2009, Topics in Geometric Topology (18.937) , notes for course 18.937. Look on Jacob's website and you will find the course notes. After that look back at the Haupvermutung book.
For the open problems I have not that long ago seen a list from the topology atlas:
I hope this helps.