B.Tech

Today, the world is moving toward digitalisation and computer science is core for it.
Today, CSE is one of the most trending branch in Industry.
You can learn more about new languages and improve your skills.
Another benefit is that most of the companies today demands coding skills either you are for Mechanical or Civil. So, it is beneficial for CSE student and helpful in getting job.

The BBDU is a very good university, providing various facilities and services for growth of student. It will be beneficial for you to take admission in it. They offers good exposure to the industry and provide an excellent opportunity to work with them. The course is explained in good pattern.

It is a good college having good infrastructure and facilities and all in budget.
It is among the Kanpur's top 5 performing colleges.
Yes, it will be good for you to take admission in Apollo college.
Apart from it you can also apply in KIT, PSIT, NARIANA etc

The process of lateral entry completely depends on availability of seat and your percentage. Top college in which you can apply are Galgotias, Amity, GLA, JSS, etc.

Being a 2nd year student you do have a lot of time to prepare as next year would largely be focused on your training and the final year on placement. Start preparing bit by bit on your own. Build a strong vocab in next few months followed by writing skills and than appear for a mock or two every month. This would ensure your continuous evaluation and would help to improve until the final day. Give your GMAT in the final year staring as this would help you with enough time to apply for the scholarships and other financial aid.

Yes, there is no age limit to study in India. The only problem you might face is the competition from the young students that are afresh and can put in a lot. Still, you may start preparing for your entrance exam and appear for the text. Best wishes.

You can go for CAT exam. It is best, as engineering students can easily crack it. so, I will suggest you to practice that.

So far my knowledge goes, you can pursue M.Sc Math in some Institute after doing B.Tech Mechanical. For example, Chennai Mathematical Institute allows B.Tech /BE for doing M.Sc Math. They also allow B.Sc Math/B.Math/B.Stat. For M.Sc Maths, you can visit their site for confirmation. Best wishes.

Try some software company for part time job

Yes, I think you should opt for PG for better opportunities and job after B.Tech IT. You can get opportunity in:
Accenture
TCS
Infosys
Cognizant
BHEL
SAIL
Tata Motors
DRDO
Bajaj
ONGC
GAIL.