Hi TE,
to my knowledge, no tools have ever been found in direct association with rock art. However, it is unlikely that the metal tools of the time (if indeed available) would have made much impact on gritstone. On some better preserved rock art panels, you can still see the 'pecking' method used to work the stone (ie; Woofa Bank on Ilkley Moor is a good example).
There are no hard and fast rules, but generally speaking, multiple rings are not common in Yorkshire rock art and are found in greater numbers in Northumbria and Scotland. Yorkshire designs tend to mostly concentrate on collections of cups, often divided or surrounded by grooves or channels and spread over a larger surface than the rings on the rock in the sample pic seem to show. There do seem to be cultural variations in design for geographic locations... but you will always find an exception to any rule you may try to formulate!
Considering the method of creation, it would be difficult to use very narrow spaces between rings, or create small compact designs, simply because it would most likely destroy the dividing rock and the rings break into each other. Although I'm sure that some creators of rock art were capable of finer work than others. Also, conditions allowing, you would expect finer carving to weather away to a greater extent than the deeper carvings.
There are collections of rock art known in the Patley Bridge area (off the top of my head, Skyreholme is nearby) but without seeing the rock in the flesh, it's really hard to say either way with any kind of certainty.
Just personal thoughts on the subject and by no means authoritative. If you want the definitive opinion on the matter, Stan Beckensall is the man.
Hope that helps in some way.
KK