a) No such thing - they are all much of a muchness and it's a complete gamble (you could end up with something worse.)
b) Every time.
c) Unnecessary.
d) Might be a good option.
"Routing" and "Wi-Fi" are nothing to do with each other. It just happens that the typical SOHO "get-you-on-the-Internet" onmi-box contains both and much more besides. (There's a block diagram of one attached to the the "Using Two Routers Together" FAQ pinned in this forum.)
So you do not need to change routers to address Wi-Fi coverage issues unless you insist on having only a single device that "does everything." (There's other caveats, but I'll omit them for brevity.)
There are no "magic" routers out there with "much better signal" than everyone else's, despite the marketing hype of the vendors - if there were, we'd all be buying them. Wi-Fi transmit power is limited by law and most kit is, and always has been, transmitting at or very near the permitted max. (It is deliberately weak by design.)
Almost always the solutions to Wi-Fi coverage or signalling issues is to get the communicating peers closer together with fewer obstructions. Since taking down the walls or asking everyone to go sit near the router is inconvenient, the solution is to deploy additional Wi-Fi Access Points (AP's) closer to where the clients are - ideally with unobstructed line of sight - creating a "cellular" coverage pattern. On big site we put up hundreds.
The "trick" with mutiple AP's is how one establishes the "backhaul" link between the outpost AP's and the rest of the (wired) network. By far the best backhaul (fastest and most reliable) is achieved using "proper" wired ethernet. Backhaul is also possible using technology such as HomePlugs, MOCA, and Wi-Fi itself, but they each have their own vices which can effect performance and reliability.
If you already have wired ethernet in situ, then it's a complete no brainer that you want to hang some additional AP's on the ends of it.
"Mesh" has no useful definition, it's more of a marketing term than anything. Some "mesh" systems can use wired backhaul, some cannot, so you need to check the specs. carefully. Some mesh systems have a "Primary" node which essentially replaces your router, but that's wholly unnecessary in your scenario and ideally you'd look for a mesh system where you can turn that feature off (sometimes it's called things like "Bridge" mode or "AP" mode.) Some mesh systems do useful stuff like pre-staging the roaming hand off, steering clients towards the "best" AP and giving you a single management platform. Again, check the specs carefully.
For an almost greenfield state where you know you will need multiple AP's, then a "mesh" or "whole home" system would seem like a good idea. But check the specs; top of my shopping list will be ability to use wired ethenet backhaul.
Such a mesh system will almost certainly not "integrate" with the AP built into (either of) your existing routers. You could either run them side by side, or turn off the routers Wi-Fi and solely use the mesh AP's system for Wi-Fi. In which case, unless the Nighthawk is doing "something else" you need, you may as well get rid of it, turn your VM router back into a router and disable the VM's Wi-Fi if you don't want to use it. Since the Nighthawk is fairly new at a time of posting, you may get a decent price selling it on which could mitigate the cost of your new fleet of AP's a bit.
Where to deploy the AP's is a conversation we can get into.