1.Is there anyway whatsoever - directly or indirectly - to learn about the path of the atom before the final screen?
Yes, via weak measurement. See the physicsworld article In praise of weakness. The wave nature of matter means it doesn't make much difference whether we're using a photon or an electron or a helium atom. For an analogy, think of normal measurement as detecting a seismic wave with a stick the size of a mountain range which mops up the whole seismic wave. Weak measurement is like using little sticks at different locations for repeated identical seismic waves. You can plot out the many-paths.
i.e. before the randomized delayed choice is made, without collapsing the wave function and eliminating the possibility of an interference?
Yes. IMHO collapsing the wave function is nothing special. There is no magic. It's something like the optical Fourier transform. Detect a photon at one slit and you convert it into something pointlike that goes through that slit only, hence no interference. Detect it at the screen and you and you convert it into something pointlike so you get a dot on the screen. It's like you're using a pointy stick.
2.Would anything interesting come from combining a delayed choice experiment like the one above and a weak measurement experiment?
I don't think so.
3.Can you perform a weak measurement, without post-selection? What kind of information would that give you? Does this help with my question 1 above?
Yes, it lets you plot the many-paths. People seem to have difficulty with this, but think of a seismic wave travelling from A to B across a plain. It isn't just the houses sitting on the AB line that shake. Houses five miles North of the AB line shake. And houses five miles south. Houses ten miles North or South shake less, and so on. So the wave doesn't just take the direct AB path, it takes many paths.
4.In the DCE setup, imagine the delayed choice not to be randomized per the paper above, but based on a predictable pattern dependent on say the time of day (odd/even time-stamp at the moment of decision dictates whether to use a screen at the last step or use the 2 individual sensors). Is there any way you could foretell the odd/even-ness of the future timestamp (at the moment of decision) before it happens?
No. There is no magic.