Variables cannot "capture non-linear relationship", because the relationship that you are talking about is a function that we are trying to learn, not-linearity is about the function, not the inputs to this function. Linear regression is able to learn a function that is linear in parameters, while more complicated machine learning algorithms can be able to learn more complicated functions.
Moreover, you can think of regression as of a very simplified special case of neural network, so if something works for regression, it should work for neural networks as well (i.e. treat them as numerical feature). Creating features based on those variables would be feature engineering and we usually use neural networks because they don't need feature engineering and learn the features by themselves.
Finally, using dummy variables for things like age is a bad idea. With using dummy variables, your algorithm needs to find individual rules for each of the levels of the dummy variable, so it finds separate rules for people who are 30 years old, and for the ones who are 31 years old. This is not the case with age, where you don't assume that people of any possible age could behave completely different then others.