Yes. A static IP assignment will remain until you change it or tell it to no longer be static.
Also, when assigning a static IP is it essential to do this both on the router and through the windows adapter settings or is only one required?
If you mean change your router's IP, no, you don't want to do that. Your router needs a fixed IP that is unique from every other device on your network - and it shouldn't change very often.
A system can get an IP automatically through a protocol called DHCP (this is what's happening when you use the "Obtain an IP address automatically" setting) - and most home routers run DHCP. Many routers support something called "reservations" where the router's DHCP can be set to always give out a specific IP whenever a system with a specific MAC address asks for one. This is setup and done on the router, and if you do this, you want to leave the system set to automatically get an address.