Good questions! There are a couple of reasons for moving the patch around from side to side. One is that after a while, the adhesives in the patches can tend to cause skin irritation, and continuing to use the same location would increase the likelihood of that happening. Once you start getting irritated from a patch, it's hard to get it to stop.
Another reason is, with prolonged use of a patch over one spot, you can tend to get vascularization in the area (little blood vessels forming under the skin to try to capture more of whatever is being released through the patch).
You can use the patch anywhere on your lower abdomen, including tummy, hips and backside... however, make sure not to rotate front to back, since absorption is about 20% higher in the back than on the front. So, once you've chosen which area (front, back or sides) you prefer, it's best to stick with that and rotate side to side.
I don't use Climara, although I have. It's a bit large, especially at the higher dosages, so it's hard after a while to find "unused" locations. I like the Vivelle dot, since it's the smallest one. Plus, it only has to stick for three and a half days, versus a week for Climara.
If you find that your patch is getting rubbed off, one thing you can try is putting a band-aid over it before engaging in any
"risky" activities. Nexcare Waterproof knee and elbow size work really well (cover the whole patch, and are very thin and clear, and stick for days). Band-Aid has just come out with a comparable product which is probably great, too.
Good luck, hope this helps!