No. Your Ac is what you see. If five minutes later it changes, it doesn't matter. If you were born at 9:00 the theme you see is the valid one. Not the one at 9:05 or 9:10.
Yes, your ascendant would be different and change your chart completely. Knowing the exact time of your birth is essential. It's a difficult situation if the hospital staff was lazy and just put down the closest round number instead of your true birth time. You can find charts where the ascendant could be 2 different signs depending on a few minutes difference in the birth time and the way to tell which one is correct is to just do two readings and see which one makes sense for your personality and life.