Should we use only one-hot-vector for LSTM input/outputs?
This depends on what your data is representing and what you want to predict.  My understanding of One-Hot-Encoding is that this should only be used for encoding of categorical features.  For example, if you have a feature representing a category of K classes, you should one hot encode this as well as the Y variable (if you are trying to predict this categorical variable).  Of course, have the final layer be a softmax to output a distribution of size K.  
This highly depends on what your data is representing.  If categorical, see above.  If just a simple numeric, you should not one hot encode this.  You could, I guess, if this set of integers is finite and small, but there is no need to learn the extra weights. You should only be using a softmax when you want to output a vector of K dimension where the entries all sum to one (perfect for representing a probability distribution over K classes).  The final layer should output whatever it is you want to predict.  If you want to predict something as simple as a numeric variable at the next time-step, just have a dense layer of size 1, with some activation function (relu probably).  More information about what exactly you are trying to predict is needed to know how to recommend anything concrete here.
I'm not sure about this.  You could have inputs and outputs represented this way, but you wouldn't use a softmax activation at the end.  You would use a dense layer outputting  a vector the size of your X variable, that is, if that's the variable you are trying to predict.  I'm not super well trained in various types of use cases for LSTMs, but from my experience, I can't think of a reason to do this.