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Having PNG format or TIFF format generated from the original files directly would be preferred at the desired resolution you need for the final product.  Capturing Screen shots might meet your needs yet goes through a color and pixel transformation to be displayed on your screen.  If you capture from the Screen, just make sure the object is large enough to conatin the desired number of pixels (or larger) than you would need in your final product.  PNG is preferred to JPEG to avoid compression artifacts.  PNG format should be fine if you prefer as either PNG or TIFF in defailt modes does not have lossy compression.

As I understand, PS Elements does not support CMYK so if you use Elements, you would need another way to convert to CMYK and to PDF files that supports the end colors you desire.  All version of Photoshop CS or CC supports CMYK

Any time you convert to JPEG the image data experiences lossy compression.  It does not recover if you subsequently save from JPEG to another non-lossy file format.

300 ppi for the final product is very high quality even for photographs.  Your image resolution of all imported objects must start at that same high resolution as well.   You cannot upsize lower resolution image to higher resolution images and retain the original quality.

I am unfamiliar with other programs and their abilities to convert RGB images to CMYK.  High quality images/colors are a matter of knowing the details of the conversion process.  If you or the publisher requires specific CMYK colors, if you don't edit in CMYK, you might need to convert and verify (and possibly modify) the colors to the exact CMYK color numbers.  This is a more involved topic and it depends on the exact specifications required by the printer.  Some want the files in a specific CMYK color space and others want to attached color space yet want the color numbers in the image to be exactly specified to get accurate color rendition.  More would need to be known about what is needed by your and the print service.  Outputting in PDF is supported by all versions of Photoshop.

Hope this at least answers some initial questions for you

Other forum members may have more inputs

John Wheeler