Well, I went into the TCP/IP settings in Win 10 and assigned a static IP (192.168.1.100), but the client list on the router says it's 192.168.1.21. But, when plugged into LAN, I'm also watching the Ethernet drop in the Network and Sharing Center. It's there, then gone, then back again, etc. This is definitely sketchy behavior, and probably explains why when I was setting it up, I kept getting kicked out and having to log back in, even when I didn't change anything. On wireless, it seems to behave just fine.
I checked the logs in the admin, and I don't see anything that looks nefarious (like frequent on/off), but it's possible I'm just not seeing it. Though really, there doesn't seem to be much of anything. I'm thinking maybe a roll back really is the best option here, just to see if I can get some stability. If yes, then there would appear to be some kind of glitch in the new firmware. If no, then perhaps the hardware is shot (even if pretty much brand new).
Unless you (or anyone else) have any other ideas, this would probably be the next best step, and I'll probably try it tomorrow, since it's getting late here and my daughter is about to get home.
Again, though, thank you very much for these ideas.