No. Anemia can occur for many reasons. If you are vegetarian, you need to make certain you get the vitamins and minerals that may be missing in your diet (if any) to prevent anemia due to lack of b vitamins, iron, etc. But this will not prevent anemia due to other causes.
I wish you could. There are hundreds of different causes of anemia, and you may get one or another. The fact that you've asked this question shows you're aware that other vegetarians promote the practice despite the danger of nutritional anemia (iron, B12). They're also risking brain damage. If you supplement, you'll keep both mind and blood safe. Be aware it's a moral rather than a health-promoting choice.
No. There are many causes of anemia. The most common is iron deficiency, and results from blood loss. This affects women significantly due to menstruation and pregnancies. A young woman who becomes a vegetarian and does not eat enough iron rich foods will become anemic eventually. However, there are many other causes of anemia that are not related to diet at all.