Re: timestamps, the app is already sandboxed and is limited in the files it can read. It's strange that they're seemingly locking down timestamps tighter than the data itself?
For free space, it's pretty common for iOS apps to restrict functionality when your device is critically low on space -- because having most disk writes fail can be very difficult to handle.
You also might want to implement a cache policy that's "greedier" when there's plenty of free space. I think there are plenty of reasons to want to know how much free space is available.
I feel like offering lower fidelity values may be a better of handling these privacy concerns rather than adding further complexity to the already onerous app review process.
For example, for files your app doesn't own, round timestamps to the nearest second, and free space to the nearest 100mb. Most legit use-cases won't need more than this level of fidelity.