Yes, you should have only src/menu/adapt-cover-menu/. Good that you deleted "box-menu-master". (FYI--If you download zips from the repo, it tacks on the name of the branch. You can feel free to delete that part of the name if you'd like.)

Since adapt-contrib-boxmenu is Adapt's core menu bundled by default, the core theme, adapt-contrib-vanilla contains styles for it. You will want to delete those, too, since some of the styles interfere with cover-menu. Delete src/theme/adapt-contrib-vanilla/less/src/boxmenu.less. 

When using the master branch of cover-menu, I was getting the endless loading gif. It was accompanied by a console error: Uncaught TypeError: could not read property 'nthChild' of undefined (adapt-cover.js line 209). In the v2 branch, this.options.nthChild has been replaced with this.model.get('_nthChild'). So I replaced cover-menu master branch with cover-menu v2 branch and reran grunt build. The course loaded and worked properly.

Optional things to do:
1. adapt-cover-menu comes with cover_menu_asset_pack. Think of this folder and the two in it as models for your design, not as a folder that should be referenced in the contentsObject.json. You didn't reference it, so I think you understand this already. However, I think you'll be happier with your efforts if you reference different images for your _indicatorGraphic. How cover-menu works will become clearer if you model your JSON like this:

"_indicatorGraphic": {
    "_isComplete": "assets/menu_btn1_complete.png",
    "_isVisited": "assets/menu_btn1_started.png",
    "_isLocked": "assets/menu_padlock.png",
    "_default": "assets/menu_btn1_notStarted.png",
    "_accessibilityEnabled": "assets/menu_btn1_notStarted.png"
}

2. My image paths above reference assets. This is a folder within the build directory. How did the images get there? As part of grunt build, grunt copies src/theme/*/assets into build/assets. It does the same thing for src/menu/*/assets if the folder exists. So I copied all the images out of the cover_menu_assets_pack folder and paster them into a new assets folder I created at src/menu/adapt-cover-menu. Then I deleted the assets pack, but you might want to save it in some other location for future design work. What's the advantage of this doing this? Primarily image organization. It's easier to copy your customized menu into other courses without having to pick the menu buttons out of the course/en/images folder.

3. After I deleted boxmenu.less, I copied src/menu/adapt-cover-menu/less/cover-menu.less and pasted it into src/theme/adapt-contrib-vanilla/less/src where boxmenu.less was. If I edit any of the styles of cover-menu, I'm going to do it in this copy, not in the original src/menu/adapt-cover-menu/less/cover-menu.less. Adapt will compile the CSS from both Less files; but if they are named exactly the same, the one found within vanilla will override the styles found within the menu's folder. This works for the Less files of components and extensions, too. It is the styles that override each other, not the files; so you could skip this step until you know you needed to make an alteration and then only include that new style in the 'override' file within vanilla. What's the advantage of this doing this? It will preserve the original Less in case you have to "start from scratch".

Hope this helps.
- Chuck