Can Image Recognition deal with classes, where the deciding quality is not directly visible?
It could and also could not. For a CNN to provide good results with no contextual input, it would mean there must be some for of correlation between the input and the output. So lets say some apps have designs that correlate with age-rating, then yes its possible, otherwise its not unless you give the network something more to work with.
This could actually set you up for a cool experiment to check yourself, run this through some run-of-the-mill CNN, and if it evaluates well (through cross-validation) then youve probably shown the correlation exists 
(Note: if the model does not test well, that is not evidence that correlation isnt there, probably isnt likely, but not gauranteed)
I have the following problem and am not quite sure if it is solvable by image recognition (and convolutional neural networks).
I have a dataset of 400k pictures divided into 5 classes. The pictures are screenshots of apps, which are put into the 5 classes depending on what age rating they received.
For example: I have 200k labeled as class 0, which means they are suitable all ages (according to the age rating); I have 50k pictures labeled as class 1 (suitable for children aged 6+) and so on.
With this data I want to train a neural network, that can tell me, which age rating a screenshot (and therefore the corresponding game) likely has.
Is this a problem, which is manageable by image recognition? 
I've looked into examples (mostly Keras tutorials) for image recognition and all of them deal with problems, which are distinctly visible (like "does the image show a cat or a dog"). Browsing through my dataset I realized, that some of the pictures are pretty similar, although belonging to different classes.
Can a convolutional neural network (or any other type of image recognition algorithm) deal with classes, where the deciding factor is not directly visible? Is this just a problem of how deep the network is?
I'd be very thankful, if someone could point me in the general direction on where to look for further information.