I can’t make a specific diagnosis on the basis of a letter. I’d need to meet with your mom. But you’ve given me enough information to confirm that, yes, she does have a problem. It’s pointless to try to reason with her because her beliefs are irrational. You and the family need to take stronger steps toward getting her help that is helpful.
The first stop should be with her physician. There are a number of medical problems that can cause the type of paranoia you describe. It’s important to check for those before pursuing a mental health intervention. It may be that the problem is as simple as a vitamin deficiency, but it might also be a more complicated medical issue.
If she is medically okay, then it’s time for a mental health evaluation. It must be very difficult for her to live with the level of anxiety and fear that you described. A combination of some medication and talk therapy could help her live more comfortably in the world.
At 17, there is a limit to what you can do. It’s unfair for the rest of the family to leave it to you to manage and protect your mom. I hope you can enlist your father and perhaps other relatives to get her real help. I suggest that the most useful first step is to hold a family meeting without her. Talk about how you can work together to find out what help is available and how to provide the incentives she needs to accept your collective support.
I wish you well.