First of all, please don't refer to any NUC using a term like NUC8i7. There are a whole slew of NUCs whose model name begins with NUC8i7. First, for kit products, you will see NUC8i7INH, NUC8i7BEH, NUC8i7BEK, NUC8i7HNK and NUC8i7HVK. Then, for system products, you will see NUC8i7INHGA, NUC8i7BEHGA, NUC8i7BEKQA, NUC8i7HNKQC and NUC8i7HVKVA.
Yes, I am being a bit **bleep** here, but you must see the issue. If you really must short-form the name, you should start by specifying the family code - IN, BE, HN or HV (all of which imply the NUC8 portion). In the case of HN and HV, the family code is usually sufficient for any discussion that does not require the specification of the system's memory and/or storage context. For IN and BE, if it really matters to the discussion (which is also fairly rare), specify the H or K for the chassis type and the i3, i5 or i7 for the processor type. For more information, consult this page: Intel NUC Identification.
So, with that out of the way, let's discuss you issue...
For the HN and HV NUCs, the Intel HD Graphics engine in the processor is not connected to nor has any control over the graphics outputs; these are all owned and controlled by the AMD HD Graphics engine. What this means is that the Intel Graphics Control Panel and Intel Graphics Control Center applications should not be used. Instead, you should be using the AMD Radeon Software application.
Finally, no, the two HDMI ports, the two DisplayPort ports and the two USB-C/TBT3 ports are all independent and can be used in any combination. Using the two HDMI ports is perfectly fine.
Hope this helps,
You can be running any version of the Intel HD Graphics driver, but the AMD Radeon HD Graphics driver needs to be the 20. 2.2 release that Intel supports (there are ways to upgrade to newer versions, but let's stick with this "supported" version for now). You can download this version from here: https://www.amd.com/en/support/kb/release-notes/rn-rad-win-20-2-2.
Now, as for the AMD Radeon Software app, unless you specifically deselect it, it is going to be installed with any version of the driver package. It is accessed by right-clicking on the display and selecting the entry for it in (at the top of) the context menu.
I have two monitors connected to the two HDMI ports on my HV, one 4K TV and one 3K monitor (no, that's not a typo, it really is a 3K monitor), and I see them both working without issue. Just to be sure, I connected these monitors to a second unit and they continue to be supported just fine.
Did you test the DRAM when you put the system together? One should always test memory accesses from a new system or a system whose DRAM is replaced. I always run MemTest86 (or MemTest86+, but that's off the table for now (forever?) now that (unsecure) Legacy Boot support is being depreciated) for a full 24 hours when I put a new system together. I consider this part of my burn-in of the system.