Dumbledore asks snape if he cares for harry then snape casts his patronus which is a doe, the same as harry mother and says always, meaning hehas always cared for harry because of his love for lily shown in the memories and the fact that he has the same patronus as her.
Draco was the one who disarmed the elder wand from dumbledore meaning it was passed on to him now. You dont have to kill to gain ownership it just must be taken against the will of its current master or something like that
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