hello,
i read through everything and this is what i think is happeneing, correct me if i am wrong, your client is basically dragging the png or the jpg that you provided him, which are at 72dpi and is asking you to fix this.
hmm first let me see if i can clear your confusion. there are basically two kinds of image files.
Vector and Raster
Vector files are resolution independent. think of Vector files as the equation of a complex line when you plot the equation you can use as many points as you want, the more points the more accurate the equation is.
some Vector file formats are .ai, .pdf, .eps etc etc.
Raster files are resolution dependent, they are the actual plot of the equation with a certain number of points. or in image terms certain number of points to describe the picture, hence DPI.
in the end the actual size (or print size) stays the same. you still print 1inch is 1inch, but depending on the input file, you might using 72 points to make that 1 inch (72dpi) or using 300 points to make that 1 inch (300dpi).
OR, you could be printing from a vector file and thats like drawing the line from the original equation, you end up printing with close to infinite DPI.
So solution, send him a Vector file like .ai , .eps or .pdf file (which have no resolution) and tell them to use that instead of a Raster file like jpg or png (which have some limited resolution)
or if they still insist on having the png or jpg files, use the export function in illustrator and export at high resolution (300dpi)
hope this helps