Some of those additional features are going to be correlated, or otherwise implicit. I think the most basic thing you’d want to do is some smoothing.

Maybe even try DFT representations, depending on how your signals behave (not familiar with EEG data myself).

In general: i depends on a lot of things.

For your specific case: it will almost certainly be better to feed it the raw data.

There were a couple points in your post that seem like fundamental (but common) misunderstandings of neural nets.

1. you cannot scale it up infinitely, that causes over fitting there is an optimal size, finding that size depends on the problem
2. summary statistics are not features. Yes it sometimes makes sense to calculate features, not it almost never makes sense to give it summary statistics
3. yes neural nets identify “features”, no they are not anything you would recognize as a feature

Deep NN do, just NN do not. In some cases the features found are not enough for good result. In some cases outperforms humans (CNN). Also, you can help the DNN by transforming the data.

The purpose of the convolutional layers of a CNN is to perform feature extraction. You can think of it in terms of classical digital processing techniques where convolutional operators have been used to extract specific features from images like edges and boundaries or correlations. The convolutional layer can be thought of as a way of learning convolutional operations to extract relevant features. Likely you will have to do some feature engineering beforehand though as you may want to preprocess by normalizing or downsampling etc, but it depends.

For a neural network, no you want to train it to represent the raw data (or near to raw like FFT) as other answers mention.

You could create a simple baseline Logistic regression model to check this. When you think about it, part of that model calculates the mean of each feature (128 electrodes * t times) as part of it's model(for binary classification). So even in this case, providing the mean isn't useful.

What would benefit from feature engineering?

If you have excel-table-style or low amount of data.

A classification Decision Tree is more likely to benefit - but this still usually only works if you can do some preprocessing with distances in embedding or other mathematical spaces, or augment the data format (e.g. appending presence of POS tags for text data, useful for logistic regression too).

A decision tree (usually) can't so easily implement things like totalling individual features, so things like totalling can on occasion be useful when you have low data numbers (although in theory, an ensemble of trees[which is the default nowadays for most] can approximate this and would if useful) - another example would be e.g. precalculating profit margin from the costs, net/gross profit for a company prediction dataset.

Std dev mean median mode entropy kurtosis and skewness

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Following this question, suppose we have a tabular data and we trained a Deep NN here. Do the weight of first layer signify the feature importance of individual features?

Give GitHub Copilot a shot and tell me whether or not AI-Assisted development can be counted as creating a feature. IMHO, It’s more of a philosophical debate over where re-inventing the wheel is occurring versus creating novel end results out of “the same old parts.”

The network only finds balance to match the inputs to the target, it will not create a feature.

A feature needs math and a purpose that you've provided; think of it as something more complicated than the starting features. For example, dividing two features while adding a third one.

The overall tendency is that AI is better at such research than human, so more work delegated to AI is better. But if your system is not very advanced and refined yet, it could benefit from additional features.