1) I think only the first two are actually stored in the PDF, but the third value (print size) is directly related to resolution (538x375) and pixel density (72ppi aka 72dpi), so it can easily be calculated anyway.
2) It seems like you're going about this a little backward. There are plenty of applications that are perfectly suited to controlling image layout and printing. Adobe Illustrator is one of the most common and there are some free ones, too. But these are going to involve loading the images, visually arranging them on the page and adjusting the print sizes visually, rather than programmatically.
2) If you did want to do this programmatically, though, I think you're going to have a hard time finding software to solve that problem. GIMP and Photoshop both have some batching capability, and I know GIMP has a fairly robust CLI, so you might be able to use that.
3) Yes, you'll start with the print size you want, divide the number of pixels by the number of inches to get ppi/dpi.
NOTE: Keep in mind that dpi goes both directions. If you have a 200 x 300 image and a 400 x 400 image, and you want them both to print 10 inches square, then you're going to distort the 200 x 300 image, stretching it horizontally. The 200 x 300 image will also look poorer quality than the 400 x 400, because you have fewer pixels to work with.
For these and other reasons, I highly recommend a visual approach, rather than a coding approach.
Good luck!