Should inputs be shuffled for training a model with expected time dependency?
You don't shuffle the actual inputs and outputs in your prediction model. Nevertheless, depending on your prediction model you might have to model the sample distribution including mean and variance. To model sample statistics you can use methods like bootstraping. However, this method assumes your model and inputs to be stationary (none changing distribution over time) and therefore neglects timely-variations, like structural breaks. But there have been modifications, e.g. bloc bootstrapping, to overcome this. You can read more about sampling in time-series here