Can a neural network recognize a screen and replicate a finite set of actions?
This is not entirely correct.
A 3-layer feedforward MLP can theoretically replicate any CONTINUOUS function.
If there are discontinuities, then you need a 4th layer.  
Since you are dealing with pixelated screens and such, you probably would need to consider a fourth layer. 
Finally, if you are looking at circular shapes, etc., than a radial basis function (RBF) network may be more suitable.
I learned, that neural networks can replicate any function. 
Normally the neural network is fed with a set of descriptors to its input neurons and then gives out a certain score at its output neuron. I want my neural network to recognize certain behaviours from a screen. Objects on the screen are already preprocessed and clearly visible, so recognition should not be a problem. 
Is it possible to use the neural network to recognize a pixelated picture of the screen and make decisions on that basis? The amount of training data would be huge of course. Is there way to teach the ANN by online supervised learning?
Edit:
Because a commenter said the programming problem would be too general: 
I would like to implement this in python first, to see if it works. If anyone could point me to a resource where i could do this online-learning thing with python, i would be grateful.