Note: I’m clearly missing something very basic here.
I have a few different documents with a difference I can’t understand 
. Some of them have a style say “Header 1” where I can edit it, and from the “Outline & List” tab change both “Outline” and “Apply List Style”, while in some others the setting is disabled (grayed).
I just don’t know what crossed relationship disables this change…
The Heading n family of styles uses an internal list style (to number your headings). To avoid any conflict or problems, interactions with this internal style are forbidden and Outline & List settings are locked.
This locking mechanism is controlled by Tools>Heading Numbering to an outline level.
If you can freely reconfigure Outline & List tab in some Heading n style, this means you have remove this style from automatic management by Tools>Heading Numbering. Since this seems to be unintentional, I suggest you revert to default configuration to avoid “surprises”.
This is not a very basic question. The disconnection is sometimes useful for special effects in documents.
PS: when asking here, always mention OS name, LO version and save format due to non-trivial differences between platforms and formats.
I’m sure it’s been me changing those settings in Tools/Heading, and forgetting about it afterward then.
What I did successfully is deriving new styles from originals, and setting for them an outline level and style. Is this a more proper/functional solution?
In my case, I always have all the headers as a numbered list.
Sorry for not specifying OS/version, I found this on Windows versions since (at least) 7.6 up to 25.2, and then I thought of it as part of the general guidelines of the application.
Until you feel really at ease with styles and other features, stick to factory configuration. Settings have been set to fulfil most users’ needs and are generally highly optimised.
This is the case for heading numbering based both on Heading n style family and combination provided by Tools>Heading Numbering.
Though this is basically just a dedicated list, it relies on a protected internal list style which cannot be used elsewhere. This guarantees consistency in heading numbering.
You can, of course, completely change the feature configuration but I’d rather recommend to reserve it for special purposes, such as alphabetically numbering appendices in addition to numeric chapter numbering. Using a non protected list style exposes you to conflicts and numbering inconsistencies because you could be tempted to apply your custom list style to paragraphs outside your outline. Moreover, if you overspecify your headings (having them both in Tools>Heading Numbering and applying a user list style), you create a real mess from which Writer is unable to escape (though this has been mitigated in “recent” releases, starting in 24.x if I am right).
Built-in Tools>Heading Numbering is a safe and good feature. Don’t reinvent the wheel when it can cover your needs.