Distributed Computing question
Beginner here, go easy.

First off, I have a repurposed mining rig (6 x 1080ti, Celeron CPU, 8GB RAM, 2 x 1200W PSUs). Second, I will be getting a Dual Xeon E5-2620 (6 cores, 12 threads ea) server with 96GB RAM, which I realize is a few years old, but it's available at my budget.

Here's my question: Can split the work so the server does the work that CPU and RAM do best, and the Nvidia rig does what GPUs do best? Or would it be better to spend my time and money finding a solution to integrate it all into one machine?
You would be better off trying to get your GPUs into the machine with those xeons actually. Assuming you can get away from risers, that is. CPU workloads are mostly pretty rare, limited to bcrypt and scrypt mainly, so you would be better served by having a host powerful enough to run those GPUs at full tilt. Cracking hashes is NOT mining, and can be a significantly harder workload for the system as a whole. You normally want at least 1 fast core per GPU, so 6 cores would be enough for 6 GPUs, and you normally want as much system RAM as combined VRAM, so 6x 11gb is 66gb of system ram minimum to support 6 1080Ti correctly, your 96gb would work well here.