Is it possible to train a neural network to learn something via video footage? Is it possible to train a neural network to learn something via video footage (which is essentially a sequence of images)?

In other words, if I have a video teaching me how to draw an animal from scratch, can I then use this video to teach the computer to draw the animal in the same way?

There are many machine learning and image processing techniques (such as RNNs) that can be applied to sequences of images or videos. So, I guess the difficult part becomes mapping the activity to an action like moving a pen or something.
Short answer: No.

Long answer: A neural network is a function that maps input data (e.g. a picture) to output data (e.g. probability that the picture contains a dog). What you propose does not seem like the sort of task this tool is suited for, though I'm not willing to claim that it can't be done.

If you are interested in computer-generated artwork, I encourage you to learn about Generative Adversarial Networks and style transfer.