Why do some of my MP3's have album cover icons and some not in Nautilus?
How can I download the album covers and view it in the .mp3 file?
Cover art in mp3 files is handled through the use of ID3 tags. It often depends on the source you got your .mp3 files from as to whether they will have cover art embedded within.
In regards to downloading cover art there are a number of programs designed to do this, some good examples are discussed in the related question: Need a MP3 ID3 tagger, and cover fetcher