Will my wiring setup work with the Nest Thermostat?
You should take a look at the Nest Compatibility Checker to be sure. Based on having G, Y, W, R, and a jumpered Rc wire, it looks like your setup is compatible. 

Note that Nest has an internal battery and does not require a C (Common) wire, but it is better to have one if you able to add it (with a currently unused wire, for example).