Hi, thanks for the detailed report, and sorry for the trouble!
Console errors: that was a bug on my side. The plugin wrongly required SGP.Systems (from another SGP plugin) even though Quest Manager doesn't need it. I've fixed it in v1.0.1. Since you already purchased it, you can re-download it for free from the project page or your itch.io Library.
Missing icon: VisuStella Main Menu Core builds the menu from its own command list, so commands added by other plugins don't get an icon automatically. To fix it, open Main Menu Core → Main Menu List and add a new entry: Symbol quest, Icon of your choice, Title Quest, and in the Handler JS put:
SceneManager.push(SGP.Quest.Scene.Scene_SGPQuest);
Then set Auto-Add to Menu to No in my plugin's parameters so it doesn't appear twice.
Let me know if it works for you!