If it's an NJM4558 from China then it's not an NJM4558, although probably a very similar Chinese chip

But I think I see an NJM4556 on your fist picture and it looks like it might be genuine. NJM4556 is beefy enough to drive most headphones, that's why Grado used it. NE5532 might work too if output current isn't too high, but it is more prone to oscillation (which would result in overheating, among other problems).
Using too weak of an opamp will not make it overheat. Quite the contrary, a too powerful one may overheat if it is driving too much power into the load. Heat is (12V-output voltage)·(output current).
Now grounding: this circuit is clearly made for a bipolar PSU and the midpoint of the PSU must be grounded. This means a PSU with three connections: ground, + and -.
If you simply apply voltage between ground and + or ground and -, it will not work.
If you simply apply voltage between + and -, the two electrolytic capacitors form a divider which splits the voltage in half and provides a ground for the circuit. But it only works for a while, because current consumption from each rail isn't 100% equal. So what happens is that the midpoint between the capacitors (ground) charges up or down until it reaches one of the rails.
Do an experiment: connect power and measure voltage from ground to + or from ground to -. You will see that one raises and the other falls.
Solutions:
- use a proper bipolar PSU
- use a "rail splitter" like TLE2426 to generate ground
- add a resistive divider (like 1kΩ/1kΩ) between the rails and connect its midpoint to ground