Distance

If your B.Tech degree is from a recognised university even if it's by distance learning, you are eligible to appear for CAT examination to seek admission in MBA in IIMs if you have minimum of 50% marks in B.Tech.

There is no procedure to get lateral entry admission in distance mode.

EMBA is good only from the knowledge point of view. There is no guarantee of placement. My suggestion would be to pursue online MBA as it will have added features of online video lectures, live streaming, problem redressal via email, etc.

Babaria institute of technology, Ahmedabad institute of technology, are two top distance learning center for Engineering in Gujarat.

To pursue MBA through distance, Sikkim Manipal University, NIBM and Symbiosis are the best choices.

Generally, Hotel Management colleges prefer students who have completed their 12th exam as a regular student. But some colleges may have no issue with the distance education. You have to check with the college directly.

Dear Kiran Kowshik,
Indira Gandhi National Open University is the top and best in the country. Secondly, you can prefer, Pondicherry Central University, Hyderabad Central University and even private institutes like Symbiosis, and Welingkar also offer distance MBA.
All the best.

No, it will be through regular board.

For searching a government job, MBA will not make any major impact as you are already a graduate engineer and meet eligibility criterion for majority of C I to Class III jobs. You may choose any specialization based on your interest and career profile.