The correct answer is no to the first question; the base metal heat affected zone will not be annealed from the heat of welding. Instead, the base metal heat affected zone will consist of an altered microstructure with mechanical properties that vary from lower to higher strength in comparison to the bulk strength properties of the unaffected A 572 base material.
Keep in mind, the altered microstructure and associated mechanical properties is a very narrow band and will be encompassed by the surrounding base material that will remain unaffected from the heat of welding.