Um, yes it is. For each object type you want to identify, you need a final probability value for that type. AlexNet works against the ImageNet data set which was 1,000 categories. AlexNet has, as its final output a 1,000 element softmax array.Um, no, that's not how it works.... Information in a neural network is not localized; it's distributed. There is no "car neuron" in HW3, sitting next to a "truck neuron", next to a "pedestrian neuron", etc
As for the number of connections, at least in the cases I'm familiar with, each layer involves a connection between each neuron in that layer and each one in the next adjacent layer.
Understanding AlexNet | Learn OpenCV
Depending on the kernel and stride (CNN), it is not a one to all mapping between layers. Even non-CNN can use subsampling between layers.