NN for fuzzy classification
I'll say it because someone else will: It depends on a lot.

Always good to start off with a regular cross-entropy loss for classification problems, especially because it comes out of the box in many libraries (PyTorch, Keras, TensorFlow).
If you're dealing with binary classification task, then the Binary Cross Entropy might be an interesting alternative to try as well.
Surely someone can recommend more obscure losses, but starting with those simple ones could potentially give a very good return for minimal effort!

Again going for simplicity, Adam optimiser or SGD with momentum is often a low hanging fruit you can try. 

I'd say use Adam and cross-entropy, and work your way through optimizing your hyper-parameters as much as you can (validation set!) and only then you can move other parts.

Also for more specific answers you should specify how many features, how many datapoints, and how you dealt with your categoricals (label encode / One-Hot).