You cannot catch an eating disorder disease. Because it is not a disease.
Its all about how much you do it. People can develop the habits that they're loved ones have if they spend enough time with them and the person does it enough. I doubt he would develop an eating disorder but may affect him in other ways.
No, not strictly but when two people spend a lot of time together, they can pick up some of the others habits and behaviour, one of the signs of attraction is that people mimic the other person subconsciously. An eating disorder comes about through unaddressed emotional difficulties and whilst they might pick up some signs of being unable to manage their emotions, it wouldn't necessarily lead to an eating disorder. If you have an eating disorder, you need to get help for it. It isn't something you just grow out of so tell your mum or go and see your doctor.
He might adapt to your habits out of sympathy and end up eating what you eat when you eat it.
Possibly. Traditionally, the man provides the food and the woman prepares it. I don't mean to put the woman in the kitchen role, nowadays you can switch it around just as easy. However, if the cooking partner prepares lots of bad food, the providing partner might suffer from the same bad nutrition.