> I think a full index of the contents of hundreds or thousands of emails and their attachments is soon going to take a lot of space and be slow on a mobile device.
The index doesn't need the attachments, does it? At a couple kilobytes per message you can fit a whole lot of text into a reasonable amount of phone storage. And there's no reason it should be slow.
> Also if you have multiple clients, which one is going to update the index and how do they sync up?
Each client can either independently index new emails as they come in or upload something like a compressed csv of new entries for the index database. A hundred new emails should only take milliseconds to process.
> if it needs to index pdfs and images.
How do you index images? Indexing pdfs is much more of a nicety than a necessity, and it could be a setting on whether you want to spend the data. It shouldn't take long though, as far as I know. You don't need to render it or anything.
> If multiple mobile clients build their own indices syncing them and merge conflicts might arise.
If they build their own then you don't need to sync.
If they share and do sync, I still don't see how you'd get merge conflicts. Emails don't change, and index updates are just adding and removing entire emails.