Is there a way to query all bounties ever set by a specific user?
If that is not possible, is there a way to query my own bounties?
You can find all bounties awarded to you and offered by you at /reputation. Bounties awarded to you are on a line that begins
I'd still like to see something like this feature request implemented, but expanded to apply to questions too.
Yes and no.
The SEDE, data dump, and API do not disclose who started the bounty (all voters except for favoriters are hidden), so you cannot use these methods.
However, as the data is publicly available by viewing the revision history, you could, in theory, screen scrape every question's revision history to see who cast the bounty. You could perhaps limit the number of questions you would have to check by (either manually or automatically) querying the reputation tab for abnormally large rep loss in a user's profile, but you will have to screen out -100 flag penalties, large amount of downvotes, and perhaps other ways to lose a great mass of reputation.