Thanks for reporting this! The log files from the two instances where Tropy crashed might be helpful; also the project file itself (just the .tpy, not the photos) if it would be OK to share it privately, but just the logs might be enough to point us in the right direction.
The same is true for any other intermittent crashes; if you can pass us the log from the crash it we might be able to fix the issue very quickly.
Apologies for the confusing access prompts; I think those happen only on the latest macOS when certain folders are accessed. In Tropy, this might happen, for example because you open the file open dialog (e.g. to import photos). It should be perfectly fine to deny Tropy access (denying access to "Photos" might have the consequence that you won't be able to browse your Photos.app library via the file dialog from within Tropy). As for the Spotlight search... do you mean the search within the file open dialog? Because that might actually be related to this, though we'd have to research this a little bit. Can you elaborate on the issue so I can try to reproduce it?
You can also read more about the permissions issue with regard to the Atom editor here. Note that Tropy uses Electron which, originally, was the application core of the Atom editor, but this issue should affect all applications on macOS when they access certain directories, it's not specific to Electron-based applications.