Lots of moving parts in there, Bud. First, the color of the cable shouldn't matter, as long as it's properly wired to the connectors. Nor is the length really an issue, as long as it is shielded, which it should be. Tell us more about he "white wire" converter. That may be part of the communications problem.
To verify connectivity, on the MBP open System Preferences, Network and in the resulting panel there should be a list of interfaces in the box on the left. There are dots beside the interfaces to show status. Green means live and functioning. Red is not in use. So on the MBP there should be a green dot beside the name of the ethernet adapter (I presume that is what the "white wire" is.) and a red dot beside the WiFi interface as it is turned off. If that is the case, you have connection to the router. If the dot is not green, then no connection. I think there is a yellow condition where it has the router but no internet, but I can't seem to find a reference. Here is all I could find at Apple:
https://support.apple.com/guide/mac-help/use-network-utility-on-mac-ntutl001/10.15/mac/10.15
Also, on the right side of the panel is a "Status" section that will tell you if it sees the router, has an IP and is active or not. Below that are the various addresses the router is issuing to the machine. The IP Address should be something in the 192.126.x.y range or maybe 10.x.x.x range. If it is any other, that probably means it is not connecting and has self assigned an IP. The Configure IPv4 box should read "Using DHCP." The Router should be at 192.168.1.1, or 10.1.1.1. The DNS server will most likely be assigned by the router, but you can manually set it on the Advanced tab.
The Wifi settings are similar, except that it also shows the Network Name of the WiFi network it is attached to. That should be the router name, although if there is another router nearby and you have not turned off WiFi at the MBP, it may have attached to that network if it didn't require a password. The way to make sure the MBP does not use WiFi is to turn it off at the MBP by either clicking on the fan shaped icon on the top bar to get the menu with the on/off selection, or in the System Preferences/Network panel, select WiFi and then turn it off there. Either way ends up at the same result. The icon on the top bar will stop showing bars for strength and show an empty fan icon for no connection.
Heat I cannot address. Not being there to see what is going on, the best suggestion is to open Activity Monitor in the Utilities folder and look at the CPU tab to see what is using the CPU so hard that it is heating up. The processes can be sorted by clicking on the "% CPU" header of the column until the heaviest user is at the top of the list. Sometimes a browser can have a runaway process, particularly if there are a lot of dynamic ads on the pages that are open, but I've never seen it get so bad the MBP overheated.
As for the "feel the the router in their bodies" comment, the signals from most WiFi routers are not very strong and should not produce any perceptible effects. The signal weakens with the square of the distance from the router, so by the time it reaches the MBP it's microvolts at best. However, there is also a transmitter in the MBP to send the signal back to the router, and the user is much closer to the MBP, so the signal strength from the MBP should be larger than the signal strength from the router, to the user. I'm not saying that they can't actually feel the effects in their bodies, but the science says that turning off the WiFi at the MBP should have a larger effect than turning off the router, or hiding it behind the couch. Typical WiFi routers have a maximum signal of 100mW output. The MBP has a similar transmitter, and although I could not find the power of that transmitter in a quick search, the WiFi standard is for symmetrical operations, so the transmitter in the MBP is probably on the same scale, 100mW. So one would think that the effect on the body would be greater from the MBP than from the router. Maybe they should not use WiFi for any purpose, including in their mobile phones, as they are even closer in use.