Would anxiety and insomnia today lead to mental health problems in future years?

Yes. If you are anxious and having trouble sleeping, your physical and emotional helath can get worse. It is better to reach out for help and reduce the anxiety and start sleeping as soon as possible to prevent this from happening. Good luck and let us know how you do.
Yes. Sleep is very necessary. If anxiety interferes with sleep, then depression may soon follow. See your pcp or a psychatrist and be evaluated.