The Crop Image action gives me LZW-compressed image data. So, yes, the image is compressed, but it’s a lossless compression. You won’t lose quality.
If the new file size is smaller than your original image (with the same amount of pixels), then this likely means that your original image was uncompressed (or less efficiently compressed).
‘Write Clipboard to File’ is working fine for me. Maybe change the target folder if you see permission problems?
You can also open images from the clipboard for example in Preview with ⌘N (“New from Clipboard”) or in GraphicConverter with ⌘J.