Distance

For high school teaching, you need to have post graduation. It also depends on what a particular school is looking for I.e the educational requirement.

No, for distance MBA there isn't entrance exam in most of the institute. If you want to do MBA from IGNOU, then you have OPENMAT. If you want to do it from Andhra Pradesh and Telangana and EAMCET and TANCET is there, but there isn't any entrance exam for Madras, Pondicherry, Kurukshetra and Bharati Vidyapeeth.

Bar Council of India does not allow students with LLB through distance learning to become their member. LLB through distance learning graduates cannot practice in Indian courts.

You are not eligible to seek admission in MBA, as Diploma is not equivalent to graduation degree. You are advised to first do graduation degree in any discipline or subject of your choice to become eligible for MBS admission.

Yes, you can.
There are many colleges which offer distance study MBA.
Sikkim Manipal University.
XLRI
IGNOU etc.

You can do class 12th with mathematics, physics and chemistry from BIOS, however, you have to do a full-time engineering degree. AICTE has banned any engineering programs through distance mode. As an alternate option, you may do AMIE which is stringent than any engineering programs.