Depends on whether you want the cover art written to the tags or just saved as a separate file to the folder.
Not touching the tags and just saving the cover art as separate files is easy: disable "Save tags to files" and configure the image file name in the cover art options.
Saving no tag but the cover art tag is more complicated. While Picard is not really designed for just changing a single tag, you can configure it to do so by telling it to preserve all tags. There are a couple of discussions here about this.
It will leave it empty. It does not actively warn you, but as it does display the cover art prominently at the lower right it is easy to spot before saving.
No. Picard can use different cover art sources, most important of course Cover Art Archive (CAA). Built-in is also support for fetching cover art from Amazon. There is also a plugin for fetching from fanart.tv. A Discogs plugin would be totally possible, but so far nobody has written one. I think most editors prefer making cover art available via CAA and copy it over from Discogs if necessary.
One last word: If you use Picard only for cover art I recommend you use the development version (also available from the Picard download page). There have been many improvements in regard to cover art handling, and the fanart.tv even requires that version.