I understand that auto-synchronizing might be difficult based on the usage of Bonjour and concerns about properly finding the last synchronized Task Coach data file.

Would it be possible to at least implement a custom URL scheme - such as taskcoach://sync - that opens the Sync screen? I could then use a timer-based tool like Workflow or Launcher to automatically pop-up directly to the Sync screen in Task Coach.

Unfortunately, I don't have a Mac so I can't modify or suggest source code. I do know that the Apple Developers page at https://developer.apple.com/library/ios/documentation/iPhone/Conceptual/iPhoneOSProgrammingGuide/Inter-AppCommunication/Inter-AppCommunication.html has details on "Registering Custom URL Schemes"

I strongly support this. Another time could be: whenever the iOS app is opened on the device, check for sync.
All of this should be, of course, with whatever computers were last used, among those that are available on the network

I see synchronisation (in my personal use) as major issue, possibly the only one