Notes not removed from index
I have removed some superfluous item notes from an item and the sync and index ran (even cleaned index) and the item notes still appear in the index (reading them in Obsidian using Zotero sync). Best I can tell, there is a bug that does not delete the item note from the index (or maybe even the record) after it is deleted from Zotero. Version 7.0.11, on MacOS 5.2 (24C101).
We can't help with Obsidian, but if it's connecting to your Zotero account online, you can look at your web library to see what's there. That's all the data that would be on our end.
In this case, I see the effect of this potential bug in Obsidian as I can delete an attached note in Zotero, let it sync and reindex (so all vestiges of the note should be gone) but it still shows in the data pulled by Obsidian plugin (Zotero sync) from Zotero.
1) The local SQLite database has nothing to do with your online library.
2) There's a local full-text index, but that's used solely for attachment content searches and has nothing to do with any visible data you see in Zotero.
3) The local full-text index certainly has nothing to do with the online library or anything that accesses your library via the web API.
Again, your items either exist in Zotero or not. If you're using some external tool that uses the web API and you see data that you don't see in the web library, that's a bug in the external tool you're using and you'll need to report the problem to the developer.