While the rear does not do as much as the front, and people do run without them, you are safer running with it. Under normal circumstances it may not make much difference, but under emergency braking, swerving to avoid something, a blow out, etc, it may well make enough difference to matter.
The ends of the sway bar should be parallel to the ground. If they are, then you need to address the issue of clearance, if they are not you should address that first and then the clearance issue.
I'm not exactly sure how they are touching the lower shock mount, do you have a picture?