There's not just one "Camera Raw Format".
What brand and model camera is the .arw file from?
a) How to explain the lower file size happening in Photoshop to the client (what is lost?)
b) Why they can use mac preview and get a larger file size that I can at full resolution jpg in Photoshop?
c) Is there a way I can make this client happy without moving to a tif file format?
d. Why is Photoshop cropping the .arw file?
a) If you are comparing an uncompressed raw file with a compressed JPEG file, it's apples & oranges. When you save a file as a JPEG, the color image data is strongly compressed while preserving the luminosity. That's what results in the smaller file size. Whether or not the color compression will impact the file down the road can't really be known. What is known is open, editing and resaving JPEGs is not encouraged. If you are supplying final full retouched JPEGs that will never undergo addition imaging, that's one thing (and ok) but if somebody down the road edits the image, and resaves as a JPEG, that's not a good thing. You might want to consider supplying TIFF files instead of JPEGs.
b) Different apps have differend JPEG encoding...again, this is apples and oranges...
c) can't tell you what your client will and won't be happy with....sorry.
d) Camera Raw/Lightroom will process raw files into RGB files. Some cameras will have automatic lens corrections applied. Without knowing the exact camera involved, nobody can really say. The question would be how are you comparing the images? What software is not showing the cropped edges?