You can specify the default file name for saving cover art at the Tag Sources dialog. Placeholders like
%artist% can be used for the file name.
MusicBrainz offers cover art in different sizes via the Cover Art Archive. The Original size can be quite large, so in addition to that, this option allows for setting a preferred size of 250, 500, or 1200 pixels.
The selected size is automatically used for all MusicBrainz tag sources.