Just eat whole eggs. Eggs are one of the most nutritional foods in the world, nothing wrong with eating them.

You are definitely underweight. Eat a caloric surplus to gain weight, eat more proteins and workout to ensure that most of the weight gain would be muscle gain instead of body fat

Anyway, BMI doesn't really tell much. It's more useful for obese people. Obesity is measure by body fat instead of weight, you can have 2 people with the same weight: Let's say 90kg. A has 35% body fat while B has 10% body fat. A is obese whereas B is perfectly fit but for both persons, using BMI show them both being obese. Get the point?