The questions are bold - my reply follows.
What is your philosophy of homework? What do you consider the role/purpose of homework?
It is practice. In a perfect world it should be individualized and time based versus number of problems based. Meaning strugglers do less problems, different problems, "easier" problems.
I also believe new material should not be practiced at home - it needs to be done under the expert (the teacher). (Comparing homework to my previous life -- homework is low level paperwork)
At the HS level I don't think homework teaches responsibility at all (I doubt it for all grade levels), it may teach a little about consequences (And again in the real world you are not punished for skipping some "homework" - skipping some low level work can be important, and for me personally it resulted in promotion -- skip good, do great)
And finally - if homework is low level - how much time should I or my students put into it? Until you reach a high level of math - my answer is not much. At high level classes with a quicker pace, it is needed that students do some learning on their own (sort of like college) but for the traditional K-8, Algebra 1, Geometry, and Algebra 2 sequence I don't think asking students to learn "themselves" is a good idea.
What are your struggles with homework?
If I assign homework - I should look at it (theoretically) but that is impossible (20 problems, 100 students is 2000 problems per day, at 8 seconds per problem is over 4 hours!). Not checking homework is my largest struggle, but checking it in class is a waste; you know 5 students watch (who got them all) and 25 are in the twilight zone. I know there is only so much of me and it is impossible to do it all - homework's return on my investment is simply not enough. I also struggle to do good differentiation on homework assignments -- that is why I am looking into more computer based practice programs (see below).
What do you think about grading or not grading homework?
What do you think? Homework is one of the hardest parts of being a math teacher - go ahead and write me a comment, or a drop me a note.