Total newbie here.
I saw that my location misses coverage, so I bought a gateway. If I see this in the console…
…does that mean a node (‘sensor’) is using my gateway?
How do I know it is a node from another TheThingsNetwork / TheThingsStack? And do these node addresses (“DevAddr”) change on a device, or is it like a MAC address?
Yes, in the sense that it transmitted some radio waves and your gateway heard them.
So you could say your gateway was eavesdropping on a device 
Gateways don’t know anything, if they hear a LoRa packet that passes the CRC check it will send it up to the LNS it is connected to. This is part of the LoRaWAN landscape.
The DevAddr. But it’s irrelevant. You just let it happen.
When a device joins or if it’s setup for ABP it gets a DevAddr from the LoRaWAN server’s pool of addresses. Many devices may share a DevAddr as they are quite short, they help the Network Server process incoming uplinks. For OTAA they will change on each join (which won’t be often) and they are not generated from anything about the device.
DevAddrs are registered in blocks to organisations so you can lookup which network a device is on.
A lot of this is covered in the Learn section (top right menu on this page) which includes a list of DevAddrs & in the docs and forum search works very well as well!