AD7616 REFCAP pin doesn't output 4V
From what I see, provided only that the supplies are correct and REFSEL is high you should get ~4.096V on REFCAP. 

So check that REFSEL is actually high (on the AD7616 side of R24), and check that there are no shorts to ground on REFCAP. You can do that by measuring the resistance across the REFCAP capacitors with power off.

Assuming the supplies are okay, then the chip must be faulty. I would guess that chances are fairly good there is a short of some kind. 



As to your other issues, that sort of thing is typically timing-related. You can troubleshoot by adding delays everywhere. Ensure you are in the correct SPI mode (a common issue). The 1K resistors will add some delay so slow down the SPI clock for testing.