I have a solar panel with below specifications. I have some questions
As in picture Vmp = 17.40V and Imp = 0.58A implies Power = 17.40*0.58 ≈ 10W.
So does it provide 0.833A (10/12) at 12 Volt ?

Absolutely not. You'll notice it will only deliver 0.66A into a short circuit under rated conditions. So if you load the output to 12 V, you'll get significantly less than 12 * 0.66 = 7.9 watts.
Bear in mind that the 'rated' conditions are just that, rated. You're unlikely to see that much illumination consistently at the best of times. The real output will be much less than this.
You are told that the maximum output is when the panel is delivering 17.4 volts. If you want the best output power, load it with a MPPT converter, and convert whatever voltage it's delivering at the maximum output power to your needed 12 V.

What is Max. System Voltage 800VDC Mean in the picture ?

These panels are often connected in series, to minimise the amount of copper wire needed for efficient connection to the load. Connect enough in series, and you get some very high voltages build up along the string. This panel's insulation is being claimed to withstand 800 V to ground. I'm not sure I'd ever run domestic panels at such a high voltage.