That means it's signed with Mozilla's signature. Actually, I think extensions can maybe be signed by Mozilla even if they're not on addons.mozilla.org.
Addons can be unlisted, ie, they don't show up on a.m.o but you can download and install them as normal.
Yes, but Mozilla will refuse to sign extensions that are not hosted in their app store.
No they wont (and don't) but it's possible they will choose to not sign these extensions, probably depending on their lawyers.
Their addon tool includes a well documented facility to sign an extension without listing it in the addon store so you can host it somewhere else. This can e.g. be used for company-internal extensions.