Are stationarity and low autocorrelation the prerequisite of regression model?
For (linear) regression models, the following assumptions have to hold:


Linear Relationship
Multivariate normality including your error term with zero mean + some infinite variance following a normal distribution
No-autocorrelation
No (strong) multicollinearity
No homoscedasticity


Non-stationarity (caused by changing mean or/and variance over time) is prevailing in most level data (e.g. stock prices) - as it has a unit root or is trend stationary. 

To cure your data from non-stationarity in most of the times, it is sufficient to use the relative change (percentage change - or log changes). Taking logarithmic changes, can also cure of homoscedasticity a bit. 

To test for stationarity, you can use the Augmented-Dickey-Fuller-Test where the Null hypothesis states that a unit-root is present in your data set, thus you have a non-stationary variable.

Coming to your second question about time-series and deep learning. Unfortunately, I haven't found a good paper/article yet that discusses the assumptions on the main data properties that have to be satisfied to get correct statistical results. See my own post - not answered yet

However, as deep learning models make use of the same underlying statistical assumptions - e.g. probability distributions etc. or even the underlying models (see e.g. linear activation function) - I guess, in a time-series framework, the same assumptions have to hold, otherwise, there is the chance of spurious regression.