how does minibatch for LSTM look like?
I think you need to distinguish between training and execution of the model. During training, you can use batches, which in your case will be different fragments from Shakespeare. So, a batch will be a list of fragments, and the language model will start from the first character on each element of the batch and do the backward and forward pass.

When you execute the model, once it is trained, you would like to see one single example, in which case you can set the batch size to one.

I believe this answers your three questions.