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.
Out of curiosity, do you have everything in /tank? As in tens of thousands of directories or files directly in the root versus a tree of directories?
Might I recommend starting without doing the router bits?
The reasoning is simple - if the rest of your homelab goes out, you just lose access to whatever you were playing with. If your homelab-router goes out, you lose Internet access.
A NAS, on the other hand, is an excellent first homelab project!
Around 90-100W typical for the part of my lab actually powered on. That is my NAS (60W typical) that has five 12/14 TB hard drives attached to a C2750 and my VM Host (35W typical, but bounces between 20W and 40W), which has a couple of SSDs attached to a R7 1700.
You don't have to run the emulators on your NAS. In fact, you probably shouldn't since your NAS is unlikely to be plugged in to any video output.
Pretty much anything can just point at your NAS's shares for access to any and all emulation. Heck, I think an Amazon Fire Stick can even do it.
If your buddies want to bring over Shield Pros to play, all you should need to do is let them on to your WiFi and have read access to your roms or ISOs directory.
So... I guess I'm not quite sure where you're going with the question?
Just keep in mind that you're going to need to make sure your hypervisor supports BIG.little architectures if you go with option one or two. I believe the latest version of Proxmox does, but just make sure you're on the right Linux kernel first.
Where are you seeing performance issues? As your issue might be less with your CPU and more with storage or some other bottleneck.