How do I learn the status of my open offers?  I can't find anything that tells me this, or even indicates that I have any offers outstanding.  And how would I cancel an open offer?  Or do open offers automatically cancel themselves after a few minutes?  As well as I can, open offers just disappear into the abyss.

When viewing a market, click on a contract to expand it and show the full details.  If you have open offers, then "Buy Offers" and/or "Sell Offers" tabs will appear next to the "Prices" tab.

You can also see a summary of offers on the dashboard if you click on the caret to expand a contract.

Offers do not expire automatically. They will remain open until they are matched, or until you remove them, or until the market ends.