On my site I keep noticing that random widgets, but more often Latest Topics, keep disappearing. I opened a ticket on this but the staff couldn't reproduce. Has anybody noticed this on their sites?

Bump.

In Mozilla console - see many safety warnings. Check them out.

The only widget I have this issue with is the System > Announcements, I have had a pending bug report for some time now on it.

Bump

I noticed two mistakes in your site.
I do not know whether those errors had an impact to your question in this topic, but it would be worthwhile to remove them.

Bump.