Can a neural network recognize a letter B as an A if your trained it so?

Does the network recognize A as an A as good as before?

Yes, it can. The reason is that if you try to illustrate the confusion matrix you can see that the FP, FN, TP and TN can be kept at a good level. What you have done is actually changing the distribution of your input data. Suppose that you train a network and it should recognize cats and dogs. You can train it just using white cats or with cats of different colours. The point here is that what you've done is not telling that $A$ and $B$ are of the same class. What you have done is classifying the first two letters as the same entity. Consider that what your network learns depends on the kind of network you use. Avoid using just Dense nets. Having image inputs, Dense nets try to find something like a mask while convolutional layers find appropriate features and those features which are better for classification would then passed to the dense layers.

Does the network recognize B as an A too?

Again yes! But it depends on the distribution of your data. It can recognize $B$s, because you have trained it in that way. But should consider that the performance depends on the data you have. How your data is obtained to be good for generalisation.