From what I can see, CS applications, fonts and font managers don't always get along. That's been my experience in the short period I have had ID. I have some versions of TT fonts used in documents that ID doesn't like even though I can see them in every other application (even PS) I have installed. And, having a large number of fonts can have an adverse effect on any application.
In looking at Font Agent Pro's web site, I see they have updated plug-ins and in one FAQ, recommend someone upgrade their FAP version for what seems like a similar issue. You don't mention the version(s) involved for Indy or FAP.
I don't use a font manager, but if I had one and was experiencing issues, I would want to know which application was causing them. I would likely begin with disabling a font manager and rebooting. I would also need to make sure that if there were fonts that more often than not caused issues, that they were in the default font folder before rebooting.
Take care, Mike