The F-ram will work BUT you have to be prepared to design the control logic to drive it. What is the full story of the device that contains this counter that you need to save the contents of ? If we knew that we may be able to find a solution by a different means to saving the contents of your counter.
Yes, if he can live with a linear progression. But I don't think that a logarithmic progression (which is what's usually used for audio) could be made to work with a binary array.
I think a motorised potentiometer type volume control would be the simplest solution for the TS as it would stay in the last position selected when powered down. I can't find just the motorised potentiometer part but there are many with the control circuits on ebay.
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the operating principle is the one in the annex.
the switch contacts shown are those of the relays. the two resistances connected to each contact are calculated to obtain an attenuation in db.
when the outputs of the 2 binary counters are all at zero the switch contacts are all closed and all the resistors are bypassed. on the contrary when the outputs are all high, all the switch contacts are open and all the resistors are used.