A couple of things I’m not sure I understand:

Is each element in the sequence a 13-d vector or just a binary value? In the second case, the input size is small and fixed, did you consider using traditional networks like MLP?
If element is a 13-d vector, how could you one-hot it in method #1? If it is a binary value, isn’t one-hot basically same as passing in the value?
What is the difference between method #2 and method#1? It seems that in both cases the inputs and outputs are binary. And cross entropy is used with softmax as usual for method #1, right?
What do you mean by promotion?
For method #3, is this an encoder-decoder model? I thought you only want to predict the next element, but it seems that the decoder is an RNN?