How are data trained in neural network?

1. Usually the input nodes are one per each feature vector, for example, if you have a 500 pixel image, and you want the features to be the pixels, you need a 500 input node network. But if you add more features, or might want to model your network with less features, you can do it as well and reduce the size of the input layer.

2. I'm not familiarized with incremental mode, but I can tell you that Newtonian methods or stochastic gradient descent may be faster than batch gradient descent.

3. Probably form a geometric point of view you can view the optimization of the NN weights as projecting your solution into the feature space, thus obtaining a concurrent vector. (same direction different magnitude)

I'll recommend you to check this course, the explanation is quite simple, and I guess you might get a better hold on how to train the NN in an easier way, specially if it's the first time you encounter them.