Not quite, this is the absolute maximum current. Sustained operation at this current will cause damage to the arduino's output pin. It says so in the data sheet.
t means, when arduino shuts down, also the relay will turn off the A/C device?
Yes that is correct.
Isnt there any relay that keeps its state (on or off) regardless of being connected to arduino/power supply
Not an electronic one. You can get mechanical relays that do this these are known as latching relays