Diploma

Dear Prem,
With BBA (IT) you can go for a certificate course in foreign language which is offered by LPU. Check the link below:
http://www.lpu.in/admission/short-term-courses.php
For more information, visit LPU camp office in your town.
http://www.lpu.in/admission/lpu-in-your-town.php#lpu-town

You have not mentioned your state from which you want to puruse your engineering Diploma course. However, if you belong to UP than you have to appear in UP Polytechnic Joint Entrance Exam (JEECUP) for getting admission in engineering and technology Diploma courses. The minimum required passing marks in 10th are 35% and if you are from Punjab then admission is purely based on the merit of qualifying examination (10th).

Yes, you are eligible and you will get a lateral entry.

You are qualified in NATA and diploma for admission to Architecture course. If you are looking for a government seat, then you may submit your marks card to Government counseling office, based on which they would give you a rank card, and you may choose colleges of your choice, college allotted will be based on your rank.

Sadly no, either you have to be qualified in NATA or IIT-JEE part 2.

Eligibility Criteria (Qualifying Exam)
Pass with 50% aggregate marks in class 10th (with Mathematics, Science and English)or equivalent.
For a candidate, who has not studied English as a subject, the condition of English may be waived off provided the candidate in his qualifying exam has studied in English medium.
Admission Criteria
Merit in qualifying examination

You can get admission but it's get upon the criteria of that specific institution. Check its guidelines if it's possible.

According to me, there is a long list of colleges in Delhi, which are providing PGDM courses. But, if we talk about the top MBA College in Delhi NCR, then we can't even pick a single name for the same. There is a list of colleges, which are competing each other to become best like:
FMS (Faculty of Management Studies
IMI (International Management Institute)
FSM (Fore School of Management)
NDIM (New Delhi Institute of Management
JIMS (Jagan Institute of Management Studies)

Dear,
Eligibility:
Candidates who have pursued Certificate are eligible for this course. Prior knowledge of spoken and written French can also be considered after validation through test.
The minimum qualifying marks may vary from one institute to another and need to be checked before submitting the admission forms to avoid end moment rejections.
Admission Procedure:
Admission criteria varies widely and interested candidates need to select the suitable institute and then need to inquire about the admission process from their offices. Admissions can be direct or through entrance test to validate the knowledge of candidates.