The diagram is too long for the list in .eps format, so I'm resending it with the diagram in .dia format. Let's hope it displays properly on your system.

On Fri, 2018-08-17 at 23:31 -0700, Samuel Sieb wrote:
The first issue is that you haven't described your network configuration 
at all.  Is this the main router that goes to your ISP device or is 
there another router/computer/device that does that?  Is this router 
connected to your network through the WAN or LAN port?
Very good questions, which I should have answered. It is not the main router to my ISP, but a secondary one. A diagram of the system is attached.

The way I configure networking is that I have a computer running Fedora 
that is the primary gateway and router for the network and then I have 
wireless routers acting only as access points connected to the network 
through the LAN ports.  No DNS, no DHCP, just connecting wireless to 
wired network.
Surely the most versatile way to set up a router is to build your own.  I had hoped to avoid the trouble and expense.

On 08/17/2018 10:55 PM, Jonathan Ryshpan wrote:
 1. What is the device android-a81a750feb8c4486?  There are only two
    devices connected to the router by wires.  Very odd.

That would be an Android phone.
Unless this router is the primary routing device, it is unlikely to see 
any devices on the wired network.
Exactly what I thought. How would an android phone appear on network management as a WIRED device, particularly as I don't remember using one or plugging one in

 2. The router has a DNS server in it.  The server doesn't know anything
    about any of these devices, so I will have to set up /etc/hosts in
    all of the computers in the local net manually.  Trendnet customer
    support has informed me that this is not a bug but a feature.

I don't understand what you are expecting here.  How would the router 
have DNS entries for the computers on your network?
The computers are plugged into the router.  See the attached diagram.

 3. There are issues with the way the router's DHCP sets up the attached
    computers so that searches aren't referred to higher level servers
    in my local net.  More devices have to be entered into /etc/hosts by
    hand.

Again, I don't know what you are expecting.  Either the router handles 
DHCP or it doesn't.  There is no passing it on.  I expect your problems 
are mostly a configuration and wiring issue.
I had thought that the DHCP communication tells computers attached to the router how to set up their routing, /etc/resolv.conf, etc. Maybe not.

Does anyone know what's going on here? Can anyone recommend a router 
that handles networking properly and that puts out a strong signal for 
my wireless hot spot?

This router appears to be supported by dd-wrt which you might want to 
look into.  I install openwrt on all the access points I use because it 
gives me so much more capability and configuration options.
I bet you're right. But (as I wrote above) I'm not interested in building a router.

Thanks for a long, prompt, and informative reply - jon