If a person thinks constantly that others are talking about her, and is very affected by what other people think of her - is this normal?

Probably not... We all have our suspicions and some may have good reasons to be so...However, excessive thoughts about this to the point of making you ill is not normal. Consult with your doc, especially a psychiatrist, who may help you sort things out. Good luck.
Maybe not. There are several problems that can lead to the symptoms you describe from depression to anxiety disorders. The first thing to do is to see a therapist who can then recommend a psychiatrist if warranted.