...Regarding the last question, are you suggesting that I use a filter on both side of the UPS?
No. One filter between the UPS and the wall should be enough.
As far as I understand it (and others here who know a lot more about this stuff can feel free to correct me here), the UPS, by design, ends up filtering the stuff plugged into it. However, the UPS itself can create noise as well. That's why you should only need one filter for the UPS.
In my case, one of my UPS-es is plugged in directly through a filter, while most of the rest of the equipment is not.
My other UPS is plugged into a power strip along with two laptop power adapters, and that strip is plugged into the filter.
It works pretty well for me. My biggest issue is that between the filters (Smarthome Filterlincs) which have the "filtered" outlet on the bottom, and the right-angle plug from some of the power strips I have, it becomes an issue to get everything plugged in properly. I wish someone made a filter with a short (1-2ft) ford at each end. That might make things easier for me. I end up having a bunch of extra power strips, and I picked up a few of those 1-ft "liberator" extension cords (designed to allow you to plug in multiple "power bricks" into one strip).