Photo Mechanic's crop is just the rectangle drawn on the image. LR by default will just show the cropped area. If you use the r shortcut in LR, then you should see the entire image with the same rectangle as shown in PM. If you only want to see the cropped area in Photo Mechanic, then go to the Settings option for the Crop tool and set the Opacity to 100%. The entire image will fill the Preview window but just the cropped area will be visible.
When you make image adjustments in LR, the updates are just saved in the XMP metadata section. To see the adjusted image in Photo Mechanic, Export a jpeg from LR. If you are working with raw files, then Photo Mechanic will pair up the raw and new jpeg, using the jpeg for both the thumbnails and large previews. Now you should see the same adjusted image in Photo Mechanic.
For your workflow steps, most people drag the selected group to LR's icon in the Dock. The Command-e shortcut does not work correctly in all situations. Many times it will just send the first selected image to Lightroom.
--Bob