I have a requirement to develop a solution based on SPFx, namely, when a document is deleted from a document library or a document is published (got main version) in a document library in SharePoint online, do something? How can I get "notification" or "getChanges" or other opportunity to handle and solve the problem?
You can't do that in SPFX. You'll have to use a web hook or a remote event receiver.
You could not achieve this with SPFx, I would suggest you create a remote event receiver in SharePoint Add-ins to acheve this: