(edit v1.4.2)
Subfolders and separators for the context menu are live in beta for this release. Search engines can be organized from a bookmark folder after enabling the option and granting the Bookmarks permission.
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Do you mean a tiered menu and separators in the context menu? I'll check it out.

(edit v1.4.1)
Simplified UI for adding engines, defaults to directly adding engines to ContextSearch and prompts for installing OpenSearch engines via the API
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The api isn't 3rd party. It's mine written specifically for this add-on. It just happens to be hosted on google's app engine with a generic name.

(edit) It probably could stand to be simplified a bit for novice users. The complexity comes from attempting to integrate with Firefox's One-Click search engines. If I kept it simple with bookmark keywords, the search engines wouldn't be added to the OpenSearch engines in the search toolbar. Adding OpenSearch engines requires an external URL, so the api is necessary. I'll look into adding a simple/advanced option that bypasses all OpenSearch integration and just adds to ContextSearch

(edit 5-11)
Sorry I didn't see your edit sooner.

If you open ContextSearch options, scroll down to Opening and Closing and you can choose a method to open the quick menu.

Automatic (popup opens after selecting text)
Mouse hold (hold either the left or right button until popup appears)
Mouse click (right-click to open )
Keyboad button (press a keyboard button of your choice to open)
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Does it not work on any website or just addons.vaultpit.io? Mozilla disables certain add-on features on addons.vaultpit.io for what I assume are security reasons. Using default settings, it should work by highlighting text and holding right-click until the menu appears ~0.25seconds.

You'll have to whitelist the native app with your antivirus software. The app is not digitally signed because it costs money to license a signature, sometimes quite a bit depending on the license length and the company used. The code is available to inspect and build yourself, or you can use the Manual method of importing your search engines which doesn't require the native app.

Hmm that's odd. It's been working for me in FF59. Would you mind answering a few questions so I can help you troubleshoot?

What OS are you using?
What website isn't working?
Is Quick Menu enable in ContextSearch options?
Have you loaded your search engines?

(edit) Good deal!

Search engines are added using the OpenSearch standard. When you visit a website and see a green circle with a + sign over the magnifying glass in the search bar in the toolbar, that indicates a website has an OpenSearch engine available to add to your Firefox search engine list (Managed from Options->Search->One Click Search Engines). Click the magnifying glass and choose Add "website". You can also visit http://mycroftproject.com/ for tons of OpenSearch templates.

(edit) I added this info to the ReadMe @ github for users unfamiliar with how Firefox handles OpenSearch engines.

(edit2) I'll look into a way of adding custom templates. I've been hesitant to implement anything like that because it would require maintaining a separate list of engines from FF's One-Click list, but it might be doable without getting too messy. Check the beta section in the coming weeks.

(edit3) v1.2.7 betas add the option 'Add Custom Search' to the right-click menu on input forms.

Oh if only it could be that simple. I could bundle the add-on with the native app, but not the other way around. Unfortunately, in order to keep the add-on hosted by Mozilla, the native app has to be separate.

(edit) I modified the native app install script (1.2.6.1+) for windows to copy the default profile path to the clipboard to make setup a bit easier

Try 1.2.4b2. Added a few options for closing plus a way to keep the menu open for clicking multiple search engines

Do you use the 'close after ? seconds' option in the other add-on? It's not difficult to add, but I can't think of a scenario where it's useful.