I'll preface this by saying that I'm not implying you are "demanding" of your family, just that having strict dietary preferences is often seen as demanding by the main food preparer of the household. For my family it was seen as almost threatening- we are quite traditional so my going veg was seen as a rebellion. Advice below is given to suggest ways to make this seem like less of a threat and less of a demand on them.
If you live with your family and rely on their cooking decisions for most of your meals, it will be hard. I went vegetarian at 14 and never asked my parents to cook based on my preferences- to use vegetable broth instead of chicken, etc. My mom is very resistant to change, so even making a simple request like that would annoy her.
I started cooking a lot on my own and sharing with the family. This means offering to cook dinner and always preparing my own b'fast and lunches. I tried to stay as integrated into the family as possible by cooking dinner that we all could enjoy- this made it a lot easier for everyone to accept my choice, and it drew less attention my vegetarian eating (i.e. this way everyone was eating the same thing at dinner, instead of everyone except me eating my mom's food and me eating a sandwich I made). From my experience, and maybe your family is different, people don't like it when one person is obviously removing themselves from family activities (including eating a meal my mom would prepare). Offering to cook the sides, or even the main at times was a useful way of ensuring food was vegetarian and not imposing too much on my family. Plus, the more my mom saw me cook vegetarian food, the more she began to offer to prepare it herself. She started making veg pasta sauce instead of meat sauce, using vegetable broth, etc. It's about slowly introducing the change without demanding anything from others. Slowly they will likely start to accommodate you because they want to, not because you ask.
Regarding your protein, I also struggle with protein. Beans are definitely the best way to get protein into your diet. Especially with mediterranean food, dishes like hummus are great for vegetarians. I'm going to venture to suggest that you can go without an egg a week without feeling sick- it's only about 110 calories and 6 grams of protein in one egg. You can easily make this up with nuts, peanut butter, fortified cereals (Kashi GoLean, etc), soy products (tofu, tempeh), or beans/lentils.
Good luck!