I realize there is probably

[gn]

I realize there is probably no way to find out if the landlord has stopped making mortgage payments…

Yes, there is. When the landlord is more than 3 months late, the lender files a NOD (notice of default) with the county courthouse. This information is public.

If you have access to http://www.foreclosure.com or http://www.foreclosureradar.com you'll be able to see this.