I'd say you can either feed the RNN's predictions into itself 13 times, or you can ask the RNN to produce 14 outputs. Not sure which of the two will produce better results, I'd say the latter if it was 3 outputs, but 14 looks like quite a lot. Anyway I don't dare to put that option in terms of probabilities -you would be putting your RNN to model $$ p({x_t:x_{t+k}} | x_{1:t-1}) $$

If not you would be cheating (as a few blog posts around there actually do...) Imagine that your last training data is for today. When you want to predict the value for today+14 days, you don't have today+13 days available yet right? However this is what you imply if you lag only for 1 day. Am I making sense?

If you want to produce a prediction for the next 14 days, your training data needs to have a lag of 14 days. However, depending on the time series to predict, it will be much much harder to get a plausible output!

Basically I agree with the answer. @Vej, please take into account that a typical toy example used in courses for CNN is the cifar10, images of 32x32, and the CNN works pretty well compared to a Multi Layer Perceptron (I guess this is what you actually mean by DNN). Cheers.

You can use regularization+cv. Take a look e.g. to this question

Have you considered association rules as a means of clustering?

Not really an answer, but only a hint: you could take a look to the package uplift. Its purpose is not the to assess the significance of the promotion, but rather to build a classifier; however some of its built-in functions might be helpful.

Take a look e.g. here. Use your 10-fold CV on the training set, as you are describing, to fix the parameters of the GBFS: max depth of the trees, learning rate, trade-off parameter and n. of iterations. Then with those parameters, take the whole training set and perform GBFS; those will be the features you select. Cheers.

Then your CV auc is actually smaller than the holdout auc right? Anyway from your description I would try to investigate what's happening with the max_depth. Even if you put something huge, your CV chooses it?

Once you've found the hyperparameters, when you say "test set", you mean some hold out data that you haven't used for CV? Thanks.