Now I have washed my hair with eggs, apple cider vinegar and baking soda for two weeks. :)
I will therefore write a little update on how it goes.
My hair feels less dry after washing it, which is nice. My dry scalp problem is still here though, but hopefully it will go away.
I have also started to rinse my hair with something sour after every hair wash. Example 5 cups water mixed with a half a lemon. This will apparently help the hairs PH balance.
I have also noticed that my hair is too long / thick for only one egg so I use two every wash.
My hair does not feel dirtier than usual (I wash as before two days per week) so I will definitely continue with this and come back with further results. :)