Thanks for posting this.
"How do you as a quality professional explain that the data provided show all parts meet specification?"
The data provided does not with any degree of confidence show that all parts meet specification because no attempt has been made to ascertain the homogeneity of the data and therefore its behavior. Sure, there are clues to what's going on, in particular the lack of agreement between the indexes C and P affecting both location and spread, this indicates a lack of predictability between the past and the present so how could this analysis be extrapolated into the future? The Process status indicates "out of control" so I assume that it means that an observation/s have breached the control limits thus further bolstering the point about homogeneity.
"What questions should be asked about this analysis to understand how to interpret the information available?"
Plot the data on a process behavior chart, if signals appear investigate and fix the issues to ensure that the data only displays random variation
"How do you answer management's question to their satisfaction?"
It would appear that judging by the histogram, the process has potentially around 40% less width than the spec so if, and only if the process is brought and kept under control and centered the company could enjoy less defects against spec on an ongoing basis.
My answer for the Key blanks is the same.