To solve its first question, WebSQL was never implemented in either Internet Explorer or Firefox ( http://diveintohtml5.info/storage.html , http://caniuse.com/#feat=sql-storage ). In terms of the "big browsers" that Chrome and Safari leave, both are born from WebKit (although, since v28 Chrome runs on fork from WebKit, it is called "Blink"). In the past, both of these browsers used SQLite as the base database for both WebSQL and IndexedDb, but Chromeed IndexedDb switched from SQLite to LevelDB.
To answer the second question, Chrome uses two different underlying database technologies:
WebSQL -> SQLite
IndexedDb β LevelDB
I suspect they support WebSQL as SQLite, as they know this works. WebSQL is now deprecated, and at some point it will be deleted, so why would they waste time porting it to LevelDB.
In terms of performance between WebSQL / IndexedDb and the performance of the base database, from experience with iOS Safari, both IndexedDb and WebSQL use the SQLite database, but they differ significantly from how the underlying database is built and how they work. In my testing, I found that WebSQL was twice as fast as 1000 simple database inserts compared to IndexedDb on Safari in iOS8.
In terms of your last question, I found this:
For IE:
WebSQL β Not Supported
IndexedDB β Extensible Storage
For Firefox:
WebSQL β Not Supported
IndexedDB -> SQLite
For Safari:
WebSQL -> SQLite
IndexedDB -> SQLite
For Chrome:
WebSQL -> SQLite
IndexedDB β LevelDB
(Sources: WebKit project, https://bugzilla.mozilla.org/show_bug.cgi?id=837141 , http://www.aaron-powell.com/posts/2012-10-05-indexeddb-storage.html )
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