Teaching & Education

Definitely, yes. You need to be at least a graduate to pursue B.Ed. So you surely apply for it. I hope this would suffice. All the best and god bless you.

No, you can also work in teaching field after BDS but the salary after it will be less.

Yes, you can apply for B.Ed.

One thing is that you can't get a permanent lectureship with only B.Tech in your hand. As per the new AICTE regulations, there are no lecture posts in engineering colleges anymore. The name of the first lectureship post is Assistant Professor. So you do need a PG (in your case M.Tech.) to enter into the lectureship. You do get lectureship as a guest in some self-financing engineering colleges or in poly techniques.
If you take your PhD, then you can start your career as an Associate Professor (Professor if there is vacancy).