I found this post for setting DPI on PNG Files. It pointed out that you should use 'metadata.mergeTree' to properly save your metadata.
With that in mind, here is some working groovy code that takes a BMP file and creates a JPG file at arbitrary DPI:
import java.awt.image.BufferedImage
import java.io.File
import java.util.Hashtable
import java.util.Map
import javax.imageio.*
import javax.imageio.stream.*
import javax.imageio.metadata.*
import javax.imageio.plugins.jpeg.*
import org.w3c.dom.*
File sourceFile = new File("sample.bmp")
File destinationFile = new File("sample.jpg")
dpi = 100
BufferedImage sourceImage = ImageIO.read(sourceFile)
ImageWriter imageWriter = ImageIO.getImageWritersBySuffix("jpeg").next();
ImageOutputStream ios = ImageIO.createImageOutputStream(destinationFile);
imageWriter.setOutput(ios);
def jpegParams = imageWriter.getDefaultWriteParam();
IIOMetadata data = imageWriter.getDefaultImageMetadata(new ImageTypeSpecifier(sourceImage), jpegParams);
Element tree = (Element)data.getAsTree("javax_imageio_jpeg_image_1.0");
Element jfif = (Element)tree.getElementsByTagName("app0JFIF").item(0);
jfif.setAttribute("Xdensity", Integer.toString(dpi));
jfif.setAttribute("Ydensity", Integer.toString(dpi));
jfif.setAttribute("resUnits", "1"); // density is dots per inch
data.mergeTree("javax_imageio_jpeg_image_1.0",tree)
// Write and clean up
imageWriter.write(data, new IIOImage(sourceImage, null, data), jpegParams);
ios.close();
imageWriter.dispose();
Worked fine for me in that OSX's Preview app and Gimp both reported that the resulting image was 100 DPI. As to Paper Size...I imagine this is directly determined by DPI? I couldn't find any JPEG property that would set that particular value.