Suggested General Reference
Principles of Condensed Matter Physics, by P. M. Chaikin & Tom Lubensky, is an excellent resource for learning soft matter physics.
It is a clear, surprisingly self-contained exposition to advanced topics in statistical physics and their applications, as well as dynamical critical phenomena, hydrodynamics, topological defects, and interface phenomena (e.g. the 'roughening transition' for solid-fluid interfaces). This is a graduate level book.
Additional general references (mainly statistical mechanics)
Entropy, Order Parameters, and Complexity, by James Sethna, is highly readable, contains many thoughtful exercises, and is free on the author's website.
Phase transitions and Renormalization Group, by Jean Zinn-Justin, gives a more concise, mathematical treatment of renormalization group methods, as well as the canonical topics of statistical field theory. This book also has many instructive examples.
Statistical Mechanics of Phase Transitions, by J. M. Yeomans, is short, but gives a great conceptual overview of theoretical techniques in the analysis of phase transitions.
The statistical mechanics textbooks by Mehran Kardar (Statistical Physics of Particles/Fields) are phenomenal. The second volume gives a comprehensive treatment of field theoretical methods, and has a nice chapter on directed polymers in random media and stochastic growth models. Both books include many interesting problems.
Polymer physics
Introduction to Path Integral Methods in Physics and Polymer Science, by F. W. Wiegel.
An introduction to standard models of polymers, and path integral methods more generally. Very well written, (but lacks exercises).
Scaling Concepts in Polymer Physics, by P. G. De Gennes,
Introduction to Polymer Dynamics, also by De Gennes.
Best Answer
EDIT: My answer assumes that you're looking for a book at the introductory graduate level.
I found Pathria's "Statistical Mechanics" (2nd ed) very helpful during my first-year graduate statistical mechanics course. Pathria's treatment of the subject is mathematically careful and detailed, at least by physics standards; I found his discussion of Liouville's theorem (part 1 of your question) satisfactory. Unfortunately, like many formal treatments, Pathria discusses few interesting applications.
"Statistical Physics of Particles" by Kardar appears to be supplanting Pathria as the favored introductory graduate text; it was used at Boston University and at Caltech during my time there. Kardar is very terse and would probably have to be supplemented by another book, but the problems he offers are interesting (if hard). In fact, about a third of the text consists of detailed solutions to the problems.
I have heard good things about Reichl's book, already mentioned in another answer. I used it briefly as a reference: the coverage of kinetic theory is more complete than in other sources. It is more accessible than Pathria, not to mention Kardar.