All of these are extremely hard to debug without actually seeing the problem, but I will try to help.
1. I only seen this happen once, there was some kind of custom code involved that caused the problem. I can only tell if I'm able to look at the code directly, to see if there is anything disabling the index table update. If you want, you can update your ticket with back-end and FTP details, and I will take a look.
As an alternative you can enable CRON updates, and make sure to have the “Index new posts upon creation?” option eneabled as well: https://i.imgur.com/JoQlHPX.png
2. Have you tried to disable the search to see if the widget content changes? If not, then it's probably unrelated, it not, then let me know.
3. Make sure that the correct custom fields are selected on the index table options panel and try re-creating the index. If nothing has changed, it should be working just like before.Best,
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