If you have the assumption that your test data and forecasting data are from the same distributions, which they should be unless data drift is occurring, then you can create the distribution of your errors when your model has been trained by looking at the errors for your test set. From there you could simply say, "This is my prediction and this is the general distribution of error, so therefore this is the distribution of my potential values," or you could take it a step further and look at the errors associated with the test data that are most similar.

There may be other techniques that work better, so further reading may help.

Can you? Sure. Should you? Not sure about that.

If you think of neural networks as underlying probability distributions, then estimating the error of your regression as single values can seem misleading, as the true error actually lies within a distribution of predicted errors. Going off of that, it would be better to analyze the distribution of your errors in relation to your predictions.