Can I connect my android phone to the LAN, via an ethernet cable?
You'll need a USB-OTG adapter, or equivalently for newer devices with USB-C ports, a USB-C to USB-A adapter, to be able to physically connect a USB Ethernet adapter to your phone.  It'll only work, though, if the phone has a driver for the Ethernet adapter you're using.  That may vary from phone to phone.

This USB Ethernet adapter has worked for me on several Google-branded Android devices: a Nexus 5, Nexus 5X, and Pixel C tablet.  I've been told by a friend that it also works on the Wii U.  It's an older device (note, doesn't support gigabit) and apparently pretty widely supported.

However, other Android device makers may have removed some drivers from the version of the OS that they ship on their devices, so there's no guarantee that it'll work with every phone.