If butter is melted then refrigerated to soften consistency will it have the proper results if recipe calls for soften butter?
No, it will not. Butter which has melted once will never again be the same after it resolidifies, so recipes assume you use butter which has never been melted. Also, when you try to warm up cold butter or also cool down warm butter quickly, the result seldom do well in baking, you get some weird behaviors like butter which smears instead of creaming well.
When the recipe needs softened butter, you should leave it on the counter at least overnight, if not longer. Shorter timespans can work, but get progressively harder to deal with, and placing the butter in an even warmer place to speed it up is counterproductive.