If his wounds are not showing signs of pus or no signs of fever, he unlikely doesn't have a bacterial infection. You can still treat him with an antibiotic if you choose to. I personally would wait unless I saw another reason to, but caution can't hurt.
While it is great that he ate 4ml of food, that is really not nearly enough food. If you can't take him into a vet and have not seen any improvement in him, I personally would tube feed him. Your duck should be eating around 30ml of food twice a day right now, and potentially more over time. Lack of nutrients can prevent his neck from healing. Even without a neck injury, they can lose the ability to hold up their head. With your ducks injuries, it is going to be hard to see the major signs that something needs to be done.
I would also only feed him mushed pellets right now. Sick ducks need balanced nutrition to really help them through injuries like this.
Also why are you putting in egg shells? At his age, high calcium levels can be detrimental to his development.
You are otherwise doing very well caring for your duck. Dealing with severe injuries for the first time is difficult for any duck owner.