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An external regressor in the mean specification can be added to the mean specification, i.e. $$r_t = \mu + \varepsilon_t + \theta x_t$$. An external regressor in the variance specification can be added to the variance specification, i.e. $$\sigma^2_t = \omega + \alpha \sigma_{t-1}^2 + \beta \varepsilon_{t-1}^2 + \theta x_{t}$$

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I am not sure I perfectly understand your question, the concept of "time series with varying density over time" is not very clear. One thing is for sure, the optimal way to "feed" a neural network is a function of the type of NNet itself and of the learning method you have chosen. For time series either you believe your data are iid vectors, and you can ...

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If you are using the "rugarch" package in R, you can include these terms via the argument external.regressors within the argument mean.model in the ugarchspec function. From CRAN: external.regressors A matrix object containing the external regressors to include in the mean equation with as many rows as will be included in the data (which is passed in the ...

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