Non-commutative space-time

Does any theory of quantum gravity dictate non-commutative geometry?
What does it mean to construct a physics on top of a non-commutative geometry?
Or, is it at all physically sensible to think of the fabric of space-time to be quantized? If time is quantized, how can we make sense of dynamics?
How can we restore the broken symmetries? Are there IR signatures of this discretization?
What about the concepts of time and space? Should we redefine them in a more abstract way?
Are we on the right track?
These are some of the questions that were added to my list of questions about quantum gravity, after discussing over one of on the back of envelope versions of arguments to non-commutative space-time with Subodh