Mining Engineering Eligibility

Yes, you're eligible to pursue mining engineering in AIT.

After class 12th, you can go for B.Tech in Mining Engineering branch. There are many institutes offering this course. ISM-Dhanbad is one of the best institutes for mining engineering.
Check the below link:
https://www.shiksha.com/b-tech/colleges/b-tech-in-mining-engineering-colleges-india

As per of my knowledge, you can't do this.
Lateral admissions are for Diploma holders.
And in 2nd year, if you score good marks, you get a chance to change your branch (if you are willing to do so).
And if the seats in MBM will be left vacant they will give an advertisement in all the newspapers informing about the vacant seat and for those seats, you can apply.
I hope it helps you.

In case of Telangana State, you can take E-CET and directly join as a lateral entry into engineering colleges.

Yes, you definitely can. You just need to take the necessary exams for the job you need.

I don't think it is recognized. I cannot suggest you this college.