Vanille-Noisette, flyfreely00 and sky princess
Vanille Noisette, the resignation letter is simply a letter that confirms that you resigned. Why would he refuse to sign that? He is legally obliged to do so after you resigned and if you request it for your new employer. There wont be a problem after all.
Flyfreely00, yes they are going to find out. Has been discussed several times here already as well.
Sky princess: the difference between OD and AD is described on their careers website andnhas been discussed on this forum several times as well. Concerning your hair, the recruiters don't care how nice your hair is, as you will have to wear it n a bun anyway for work. Just make sure you look neat and business attire-like.