FYI: Your imgur link is invalid.
Yes, the GTX 1070 should work, assuming it has two 6 pin PCIe connectors. I assume it does because you already have a GTX 970 in it, but I can't find detailed specs of that power supply to confirm.
As for the option of swapping out the power supply in the future, I'm not sure that you can. I found a few message board posts from individuals stating that the pinout on that power supply in a Dell Precision T5400 is non-standard. Is this is true, it would mean that a T5400 isn't compatible with a standard ATX power supplies; pins in motherboard cable have different layout. I do not know if this is the same issue for the Aurora R3. They appear to use the same power supply.
I would recommend that you purchase the GTX 1070 from a site that has a no hassle return policy (e.g. Amazon, NewEgg, etc). If you ever decide to upgrade the power supply, then I suggest that you search YouTube for people who've done it, and ask them if they needed a ATX converter cable to install a new power supply. And if so, where precisely did they obtain it from, would be the question you should pose to them in the comments.