Thanks for using Morning Coffee Quantum! Let me see if I can help with the issues you've mentioned.

You can probably get the icon back via Firefox's Customize feature. In the toolbar to the right of the location text box, look for an icon that looks like three horizontal lines. Click it, and you should get a menu that has a Customize option about halfway down. Click that, and you (hopefully) will see the Morning Coffee Quantum icon among all of the available icons. You can then drag and drop it back onto the toolbar.

Unfortunately, due to the way the new web extensions work, I couldn't set up a custom right-click on the button. But you can add sites to your daily lists another way - through the tab context menu. After you've opened a tab for a site you want to add, right-click its tab and select the "Add Tab to..." menu item in the menu that appears. Then you can pick your day(s) to which you want the site added.

Once you've added your sites, clicking the coffee cup icon should load them just like you remember. I hope this helps!
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In response to your additional question: the add-on doesn't have a "open your daily sites upon launching Firefox" feature yet. I might try adding one in the future, if there is enough interest and there exists the capability to do so. Currently, you have to click the button (or use the key shortcut) to open your daily sites. On the options page, there is an option to close all of your currently-open tabs, replacing them with your daily sites, when you click the button. That might get you something pretty close to what you want.