Lateral entry

Yes, you can take admission for lateral entry. Many colleges in India offer lateral entry for BE. Some of them have entrance exam while some give direct admission. Minimun eligibility for any college is you should have passed Diploma with at least 50%.

Chances are less for BBA after Diploma, but I will suggest you that go for B.Tech in civil engineering, because as compared to BBA, Civil is much better because after B.Tech in civil engineering, the job prospects are more as well, as through your B.Tech degree, you can get PWD license, and by that you can set your future. So, choice is yours.

Well, both the colleges have lot of such opportunities. Personally, since I have been to many fests and workshops like UTSAV and technical conferences, BMSCE definitely has quite much.

General MCA, you will have a full time 3 years course but Lateral entry will take your 2 years only. After admission in MCA with lateral entry.
You have to give first year exams also with your 2nd year exams. After completing these two you have same value of degree.

Hi,
You may do the Diploma in engineering and then enter second year for engineering.

CUT off marks for recent years are available at Shiksha website or app or data is available at college or university website or JEE Main website, you can check there for better results.

No, the lateral entry is not possible in IIT and NIT, it is only possible In regional government and national private college.

As far as my knowledge is concerned, there is no lateral entry (directly begin from 2nd year) in B.Tech AMU. Check the eligibility criteria for the entrance test here:
Senior Secondary School Certificate Examination of this University or an equivalent examination with at least 50% marks in aggregate of English, Physics, Chemistry and Mathematics.
Or,
Diploma in Engineering of this University with at least 50% marks in aggregate.