I guess let me first say that FOG is intended for Workstation class computers not servers. Typically servers have specialty hardware that requires unique drivers which are not common in the standard linux kernel.
With that said it looks like FOS has the right drivers for the DL580 because it is seeing the network interfaces (Yes!!).
Your network interfaces are being seeing by the iPXE kernel because you are getting into the FOG iPXE menu, it just appears that FOS (customized linux OS that captures and deploys images) is having a hard time connecting to the network.
In one of your pictures, the condition reminds me of a networking issue with Spanning Tree protocol, in that you have the default spanning tree protocol enabled and you are not using one of the fast protocols (RSTP, MSTP, fast-STP, or what ever your switch manufacturer calls it). An easy test for a spanning tree issue is to place a dumb (unmanaged) switch between the pxe booting computer and the building network switch. Then pxe boot.
Lets try this before we move on to the next step.
Oh, one thing I noticed from your second picture is that FOG is sending the 32bit kernels to this DL580, I find that a bit odd since this is surely a pure x64 bit system.
Lastly, the pictures are great because they explain the context of the errors very clearly, well done!