1. Ads never expire.
This is totally a cron setup here. Have you tried running sendnotice.php in your browser to test that it expires ads?
Cron Job: sendnotice.php
2. Classifieds will take someones payment for an ad but not post their ad, leaving me with many emails and refunds to issue.
Well do you have approval or moderation as it is called set to yes? My suggestion regarding payment is if your running an old classifieds try out the new 3.0 which has payment verification through paypals ipn service which posts back the data to ensure successful payment before it relied on a user clicking the return link so what happens here in older classifieds if the user payed you and did not click the return link the ad is never approved. To avoid this upgrade to 3.0
3. Auction doesn't work - never has.
It should work if you allow in edit categories bidding on ads within that category and the user hits to allow bidding. Again the auction needs the cron task run to expire the auction so your issue might be tied into not running the cron task. You may need to talk with your host on how to setup and run a cron task on your server.
4. Classifieds will send up to 10 notifications for a single ad upload. This has been an issue from the get go..
This is indeed an old bug and has been fixed in later versions so upgrading to 3.0 should help you.
Photopost Classifieds 3.0 Released 3-09-08