"Local Storage" folder is for HTML5 window.localStorage
"Local Extension Settings" is for chrome.storage.local.
> This command gives 0 bytes even when the extension has a leveldb log file in 'Local Extension Setting' directory.
Assuming you ran getBytesInUse in devtools for the extension's context, the log file may be present but be empty inside e.g. if the data was written and then deleted. Chrome postpones compaction of the database, so it can retain its size for a [short] while even if there's nothing inside.
> unlimitedStorage removes the storage quota for the localStorage
No, unlimitedStorage is applied only to chrome.storage.local as noted above.
HTML5 localStorage is always limited to 5MB (computed as a combined string length which actually means 10MB physical bytes on disk because strings are 2-byte unicode in JS).
IndexedDB, sandboxed HTML5 FileSystem use the global quota manager so they can grow a lot: