How does ML algorithms treat unseen data, a conceptual discussion I want to predict the occurrence of certain events, but these events only occur say 5% of the time in my data, hence in 95% of the data there is nothing to learn. 

In order to teach the ML algo something I have learned to single out the 5% and drop the rest of the data. Let us say that I want to predict if a picture is of a dog or a cat. In my data 2.5% of the pictures are of dogs and 2.5% of cats, the rest are just random pictures. So, I single out the cat and dog pictures and label them so that the ML algo can learn from that. Am I broadly right so far? 

So, if I train my algo on only cat and dog pictures and get a satisfactory accuracy, what will then happen in live usage when 95% of the pictures are not of cats or dogs? I.e. I show my model a picture of a house, what does it predict? Will my algo always predict either cat or dog, or will it somehow tell me that it has no clue what this picture is? 

Any thoughts?
Define two flag variables: flag_is_cat and flag_is_dog, which take on values of 1 if the picture shows a cat or dog, respectively, and 0 otherwise. Define another flag that takes on the value of 1 if the picture contains either a cat or a dog. In a word, label the data.

If you train the model using all of the pictures, even those with neither a cat nor a dog, then the model outputs a probability that the picture contains a cat, a probability that it contains a dog, and another that it contains either. This is the approach mentioned by @marco_gorelli . Dividing the probability that the picture has a cat by the probability that it has either a cat or a dog gives the probability that the picture has a cat conditional on the picture having at least one of them.

Alternatively, if you train a model using only those pictures that contain either a cat or a dog, then the model would output the probability that a cat is contained in the picture and that a dog is contained in the picture conditional on at least one of them being in the picture.