My child has been diagnosed with autism: 1st--can she get better? 2nd--will she lead a normal life when older?

Can only talk odds. Without even knowing your child's age or severity of symptoms, any answer is going to be highly general. Yes, some children with autism get better, sometimes markedly so; however, most continue to have problems lifelong. Usually, even with children with no problems, knowing what kind of life they will lead is impossible to guess until they make it through their teens.
Perhaps it would. Be helpful for you to rent the movie "temple grandin". Let me know what you think.
She will improve. The path followed by kids bearing an autism or autism spectrum disorder DX is different from average kids but improvement is almost always seen.Since each kid seems to differ, even with others carrying the same label, it is hard to plan ahead for how quickly they improve. Recent studies show rapid improvement in some even within a year. Lifestyle when older depends on the degree of improvement.
Yes, New Normal. Have hope that your child will get better with intensive therapy, social skills training and support. Her life when she is older will be what I call a " new normal" You will grieve for the loss of that "perfect child" ( which none of us have, by the way) Then you will work towards building the best life possible for her! Much of this depends on severity of her disorder "on the spectrum."
Yes. The answer to your first question is yes! The earlier that she can get therapy for her autism the better! People with autism are capable of amazing things, and only time will tell how far she will go! Temple Grandin went on to get a Masters Degree! See this link for more inspiring people with autism! http://www.autismspeaks.org/news/news-item/10-talented-individuals-autism-who-inspired-2014.