Eating DO. Sure, same as in anxiety leading to social avoidance and withdrawal due to low esteem and fear of rejection.
Sure. Dating is stressful enough. Eating disorders add an extra amount of stress, especially about eating with other people, letting them know about the "secret", whether it is restrictive (anorexia nervosa) with or without purging (bulimia nervosa). Just another of the adverse effects of these disorders; aside from the physical issues, they can cause a lot of emotional and psychological pain.
Not likely. Eating disorders do not cause any problems with dating by themselves. Anxieties about dating or going out to dinner may result from one's feelings regarding the date and being in certain social situations as well as the person's emotional and psychological state.