Que 1. For how many epochs and for what batch size(lets say for mnist
digits dataset) do I need to train my model?
This depends on your model. Different models may perform differently when trained on different number of epochs, with different batch sizes. Basically, batch size, learning rate, dropout and other regularization methods all have some regularizing effect and interact with each other. They also interact with number of epochs needed to train the model (more regularization needs more training epochs).
Que 2. Is there any time condition to train my model, or I can train
my model for same no of epochs as other models? Like lets say
comparisons have been made after training models for 1 hour. Or is
there any condition that I have to train my model for particular
amount of time(like 1 hour)?
No, unless you want to make comparison under time restriction. Notice that this would need you to re-run all the other models in the same conditions as your model, since the results will vary a lot depending on the computational resources that were used by different authors. If you won't re-train the models, you wouldn't know if the results are due to having better (or worse) computational resources or using better model.
Que 3. What is the state of the art accuracy and error for mnist
You can find such results (with references) on LeCun's page on MINST or here.
Que 4. How do we calculate error?
Error rate is 1-accuracy, but accuracy is not the best measure of performance, so you should consider other measures as well.
As a comment, MINST is not a great benchmark since everyone trains on it, we probably have NN architectures that overfit to MINST.