A router's detection of a client can sometimes be delayed or buggy. The ethernet interface of the Pi should disappear from the router's client list after it's been disconnected for some time, and if it does not, then a reboot of the router should correct that issue. It's not a problem for it to simply be listed, but I understand your frustration with it still showing while disconnected.
You can set the static IP for the Pi on the router or the Pi. If set on the Pi's end, the Pi will not request an IP address from your router and will force its interface to use the IP you specify. If you set it on the router's end, the Pi will still request a dynamic IP from the router using DHCP. But, the router will always assign the Pi the same IP address that you set. It's up to you how you wish to deal with it.
Without knowing your router model, I can't provide you with specific instructions on how to set a device's IP address, but most routers have a "DHCP Reservation" option that you will want to set. As for setting it from the Pi itself, that can vary too, but what it comes down to is that you need to follow the instructions for the OS you are running, which I assume is Raspbian.