Does hunting or slaughtering birds require prior intention, and if the bird is found alive after being wounded, is it obligatory to pronounce the Name of Allah upon it again before slaughtering it?

You are not required to intend that your hunt is for Allah, but rather it is stipulated that it should not be for other than Allah Almighty, and that you mention the name of Allah over it. For hunting to be permissible, there are four conditions: the hunter must be a Muslim or a person of the Book, the hunting tool must be sharp or the predatory animal must be trained, the hunter must intend to hunt when shooting, and the name of Allah must be mentioned when releasing the arrow or the predatory animal. If these conditions are met, the hunt is permissible, even if you did not intend to draw closer to Allah through it. If you find the hunted animal alive with a stable life, it must be ritually slaughtered. However, if its life is unstable, or there is not enough time to slaughter it, then it becomes permissible solely by hunting.