What is the best Keras model for multi-class classification?
Your choices of activation='softmax' in the last layer and compile choice of loss='categorical_crossentropy' are good for a model to predict multiple mutually-exclusive classes.

Regarding more general choices, there is rarely a "right" way to construct the architecture. Instead that should be something you test with different meta-params (such as layer sizes, number of layers, amount of drop-out), and should be results-driven (including any limits you might have on resource use for training time/memory use etc). 

Use a cross-validation set to help choose a suitable architecture. Once done, to get a more accurate measure of your model's general performance, you should use a separate test set. Data held out from your training set separate to the CV set should be used for this. A reasonable split might be 60/20/20 train/cv/test, depending on how much data you have, and how much you need to report an accurate final figure.

For Question #2, you can either just have two outputs with a softmax final similar to now, or you can have final layer with one output, activation='sigmoid' and loss='binary_crossentropy'.

Purely from a gut feel from what might work with this data, I would suggest trying with 'tanh' or 'sigmoid' activations in the hidden layer, instead of 'relu', and I would also suggest increasing the number of hidden neurons (e.g. 100) and reducing the amount of dropout (e.g. 0.2). Caveat: Gut feeling on neural network architecture is not scientific. Try it, and test it.