Teacher

You can be a teacher in elementary level with D.El.Ed. If you want to be a TGT, you need to complete BA and B.Ed. If you want to be a PGT, you need to complete MA and B.Ed.

If you are a graduate then BEd is a good course as it will help you to get TGT post teacher jobs. Once you obtain Post graduate degree with BEd you may be considered for PGT jobs.

One you achieve minimum eligibility criteria for Nursery or primary or TGT or PGT or College or University teacher, your gaps years will not impact you search of teaching job.

Well, for this year, the central government has no such plans to add any new colleges for B.Ed.
There are no plans by the central government to offer B.Ed for free.

Yes, IGNOU provides that opportunity.

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

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).

You need to pursue B.Ed after completing UG.

No, you cannot. BA and B.Ed are compulsory for TGT.