Zenphoto does not "fail" sporadically like that, it's simply not possible; something changed outside it to affect the environment it was running under.
It could be an .htaccess file in a higher directory... check those.
Have we tried re-installing Zenphoto yet? Especially from the latest version? Please try that, it seems like you have an old version...
That aside, I've narrowed it down to a problem with getting the album thumbnail image... the first album works because you've selected a thumbnail for it manually. If you can, do that for the others and see if that works.
If you can get to your server through FTP or SSH (Shell) make sure the permissions on the /albums directory *and all albums within it* are readable by the web server (generally permissions of 755 for individual folders, with 777 for the /albums folder itself). It's also important that the server has execute permissions (hence the '5' in the last place) for each folder folder, so it may list the folder contents.
Sounds like you're generally just having permissions problems. The server needs to be able to read the directories and list their contents. Could be that the permissions got changed somehow, that's not uncommon.
You need not worry about zenphoto "failing some time in the future." That's really not possible unless you change something such as those permissions, even accidentally.
Also, I wouldn't worry about Zenphoto being hacked. It's quite secure.