As far as I can tell, you can achieve what you want by doing this. Click the button at the top right of the main panel, above the scroll bar and below the column browser. From there, go to Sort By>Define Custom Sorting. In that menu, go to the grouping section and change "The following fields..." bit to 1) Artist, 2) Album Name.
Be warned - if you have multiple tracks from one artist with no defined album tag, this will still produce strange things. This is not MB's fault - it's designed (like all music players) for properly tagged files, and it's doing the best it can with ones missing the basic tags (namely track number, artist, album and album artist).
As for the searching problem, MB searches all fields - it's entirely possible that a comment field, or even one not visible in the main tag editor, contains the search term. To check, open the offending file in the tag editor (floating window) then click on the Tag Inspector button. That shows you all the tags on the file, and you can delete any you don't want.
Finally, it's possible you've got a setting enabled that's giving you problems in artwork view when searching. Go to the artwork tab, click on the same button as described in the first paragraph, and go to Search Results. If 'Always Show Entire Album' is ticked, untick it.
Bold words in my posts are links unless expressly stated otherwise.