With regards to your questions.
1) Yup, see Example 9.3.31 and Theorem 9.3.37 in Lazarsfeld's book, "Positivity in Algebraic Geometry II".
2) You might not be aware of things like (minimal) log discrepancies. Of course there are various properties of the minimal resolutions too.
3) Exponents Do you mean exponents of different variables , or the same variables, what did you have in mind? I don't know of any good geometric interpretation of doing this in general.
4) This log canonical threshold also appears in many other unexpected contexts? See for example:
Budur's initial question in "Singularity invariants related to Milnor fibers: survey". http://arxiv.org/pdf/1012.3150