The GUI way would be to select all files in which you want to delete the tags, presst Alt-T to get the extended tags dialogue and then delete there all fields one by one.
As the cover is treated separately, you should be left with the cover afterwards.
You are on the right track: cut the tags and then paste them immediately again.
The "Lyrics 2.3" tag is a rather awkward tag that can be removed safely and this is done if you cut and paste the tags with the function of the context menu (right click) in the files list.
THis won't delete your unsyncedlyrics tag but clean up your tags that in the end you have nothing but V2 tags.
Of course can you hide that column - right-click on the column header, select "columns ..." and untick the appropriate column in the list of columns ....
Yet, that column shows the kind of tags that are present in the file. So hiding the column does not modify the file contents.
One way of modifying the tag version (of which "Lyrics" is one) you have to cut and paste them as described before. If you cut and paste the tags before you deleted the cover and set the options for writing tags to "V2.3" then this should restore the cover if you paste the cut data immediately.