Did you mean an i5-8400 instead? As there is no i5-8700 - that’s an i7, which would have the same number of cores as the 9700.
The i5-8400 would be fine for most IT homelabs; there are some specific applications that really need more power than that (especially any of the Cisco bits from what I’ve heard, but I have no first-hand experience with those).
What exactly are you planning on doing with the homelab?
For HP, you’ll see a model like an HP Elitedesk mini 800 G4, for instance. Elitedesk will either come in a 705 or 800 - this is just AMD or Intel. Prodesk will come in 400s or 600s - this is mostly chassis and cooling for differences, with anything ending in a 5 being an AMD model. The “G4” part is the HP generation number. A G4 corresponds to a 7th or 8th gen Intel or a Zen1 CPU (or Celeron/Athlon equivalents on the low end). These numbers line up with their SFF and full fat desktop brethren, for reference.
All of these machines can come with 35W, 65W, and 95W TDP CPUs (or closest to those numbers in the case of the AMD CPUs). If you are looking at the minis, you’ll want to avoid the upper end (which are all K-series CPUs) on account of the fan needing to constantly run to cool them. Usually a T-series (or E-series for AMD) CPU will have the same power draw as a non-T CPU, but lower performance and lower peak heat production, which is actually important here because of how small these boxes are.
Depending on how old you go, the minis mostly have either an M.2 slot plus a proprietary port for SATA (2.5 inch drives only) or two M.2 slots. There is a configurable portion the back that can be anything from a VGA/D-SUB out to a Thunderbolt port or even a full dGPU (which would also use up the space for your SATA drive). If you buy off-lease hardware, it is almost always the former, but swapping these out are super simple. Just make sure the part exists for the exact model of miniPC you want - they get stupid specific on what is and isn’t compatible.
Depends on your gaming box.
If your gaming box has fewer cores (say, an older i5 or current i3 CPU), you might notice Nextcloud running. Pihole is so light I doubt you’d notice even if your gaming CPU was a potato.
If your gaming box is more than four cores, I doubt you’d ever notice.
Where are you seeing performance issues? As your issue might be less with your CPU and more with storage or some other bottleneck.