Is the 1st-year of MBA same for all MBA graduates irrespective of the specialization?

Yes, the first year for all specialisation is same in MBA. If there is trimester pattern then 1st 2 trimesters are same and from 3rd trimester you have the specialisation subjects.

Yes, first year of MBA is same irrespective of specialisation. First year courses covers topics like Accounting, Economics, Corporate finance, Strategy/Frameworks, Human Resource Management, Organisational Behaviour, Leadership, Marketing and Information Technology.