This is an issue unrelated to this addon. But If you want to save the width adjustment of N Panel, you can save the Startup File after adjusting the width of N Panel. The menu path: File -> Defaults -> Save Startup File.

No, it doesn't work with other N Panel manage addons. You have to make sure only one of these addon updates your N Panel, otherwise, they will lead to chaos. If you need to use another N Panel manage addon, you can just uncheck the Auto Apply On Startup option of this addon and restart Blender and not apply the settings manually. Thus this addon will do nothing to your N Panel, except for the HardOps and BoxCutter addons which may need you to modify their tab names in their addon preferences. Then you can use another N Panel manage addon. BTW, Be careful if you disable this addon, it will lose your settings. Before disabling, remember to export the settings, or confirm that they are useless or can be easily rebuilt.

In most cases, you may have modified some categories settings, but not applied them yet.  Before you search for an answer to an issue, you should confirm that the categories settings have been applied. That's because they are already associated with the N panel after your first applying. If you change the categories settings, the association will not be completely correct. Then some weird behavior may appear. So you need to apply the categories settings after you change them.

They are some kind of special. If you want to manage these tabs like other tabs, it may not work as you expected. It may be a bug or a decision of the Blender. So in this addon, these tabs are in the Exclude Tabs, and they are not allowed to be deleted. If you meet any other tab that doesn't work with this addon, you also need to put it in the Exclude Tabs.

You may have recently removed/disabled an addon, but that addon did not completely clean up(i.e. unregister) its panels. You just need to restart Blender to clean up these panels. In a future version, this kind of panel may be filtered directly. Edit: From version 1.1.0, this kind of panel is filtered directly. No longer need to restart Blender when you remove/disable such addons.

This addon works on the N Panels of most addons on the market. But there are a few addons that have their own styled management on their N Panels. This may lead to an error. If you found such an addon, you can add their related tabs to the Exclude Tabs to avoid bothering, and report it to us. We will try to fix it, but not a promise. The other way, you can report the issue to its creators.

The first method is to uncheck the Auto Apply On Startup option in the addon preferences and then restart the Blender. Thus this addon will do nothing to your N Panels, except for the HardOps and BoxCutter addons which may need you to modify their tab names in their addon preferences. The second method is to disable this addon and then restart the Blender. Note that the second method will lose the settings. If the settings are important, please export and save them first.

First case, you may have disabled the addon, but the tab is still in the category. Second case, the panels of that tab need to meet certain conditions before they are shown. For example, you need to select an object, or the editor needs to be in a certain mode.

It's possible that you changed the category settings and it wasn't applied, or you mis-dragged the order of the panels, or for some unknown reason. You can re-apply the category settings, or correct the order manually by dragging the eight-dot icon at the top right of a panel. By the way, if you change the category settings, remember to apply the category settings to update the sidebar.

That's because some addons use the bl_order to control the category order in the sidebar. You don't need to worry about this, it is not a bug. If you don't like this, you can put it in one of your categories, or change the bl_order in their code.

If you update Exclude Tabs, you need to restart Blender.

No, it only works on the 3D View N Panel. 

Yes, but the tab's panels only show in the last category in which you enable it. 

It is a temporary category where all panels of disabled tabs in the categories are temporarily stored. Normally you don't need to pay attention to it.