Distance

Definitely, NIFT is the best institution for a design course.

You should opt to pursue the same through regular, as HM is a practical course, therefore, the exposure will be limited in distance courses.

B-Schools do not use CAT or MAT score for MBA by distance learning. Every B-School or university have its own process to select candidates for MBA distance learning. e.g. IGNOU conducts it's own entrance exam for MBA distance learning entrance examination.

Both are good.
PGDM is an Institute autonomous program which is approved by AICTE
MBA is a program which is affiliated to a University.
Colleges which offer AICTE approved courses will also offer PGDM not MBA, for example XYZ College is affiliated to ABC University they offer MBA course affiliated to t

No, you can pursue either B.Sc or B.Ed at same time, you cannot do simultaneously.

Yes, why not, this is absolutely possible.