You should refer to ASME VIII Div.1 UCS-66 to see if impact test is required (Curve D for SA-516 Grade 70 Normalized)
Generally carbon steel is no "adequate" for temperatures below -29ºC (-20ºF), but SA-516 if normalized has good working below that temperature.
For example, if temperature is -48ºC(-55ºF) or less you will need to make impact test for any thickness you have.
For other temparutes it depends of the thicknes. Please, have a look in the code.
Take note that independtly of the material and impact test you have for shell and heads (assuming you are talking about a pressure vessel...) you must considere material used for nozzles and flanges.
It depends on the application as well. The ASME B&PV Code has specifications on when plate may be used at various temperatures, but the various tank standards have differing requirements.