M.A

Well i hope should get admission into MA. However, it is advisable to go through the eligibility criteria of the University itself.

No.M.A and M.Ed are two different courses.At first complete Your B.Ed degree without which you can't apply on teaching jobs. After completing B.Ed you can complete Your M.A or M.Ed degree. According to my view there is no need of M.Ed degree to get a teaching job but M.A is important.it has an academic background.

As per my knowledge Patna University is better. However, you need to check with the faculties and with the students enrolled. It will help you grasp the actual scenario.

MA is a normal MA, as the syllabus is normal and you can find Guide and books easily for almost every subject.
MA (hons) it tougher then normal MA, as it have different syllabus and guides and books are not available for complete subject, even some book carry only 1 chapter in a book. so it become harder for student to complete course in given time. it is like researching on a subject.
But as in India when we talk about job, we like to recruit that person who had spend more time in study then other. so MA (Hons) get more preference in job then simple MA guy.

Yes, you are eligible for NET as you meet the eligibility criteria.