Why do we have $I_+= I_- = 0$? Does it have anything to do with $I_G=0$ in a FET?

Well, yes, sort of (note that the transistors inside op-amps can be bipolars rather than FETs-- that's the case for the vanilla 741 op-amp). The op-amp inputs do indeed lead into transistor bases, and current drawn is very small (they have high input impedance).