Ask & Answer: India's Largest Education Community

Yes, 12th is mandatory for NCHMCT JEE. The eligibility criteria for NCHM JEE is given below.
Eligibility criteria: 10+2 system of Senior Secondary examination or its equivalent with English as one of the subjects. Candidate must have passed English as a subject of study (core/elective/functional) in the qualifying examination. Those appearing in 10+2 or equivalent examination can also appear in NCHM JEE 2020 on provisional basis. Provisional admission will stand cancelled if proof of having passed the qualifying examination (10+2 or its equivalent) is not submitted at the time of counselling or at the time of admission or the latest by 30.09.2020.

Yes, improvement exam will be conducted among the students who are not satisfied with their marks and wants to improve their marks of 12th. Students can enrolls for subjects in which they wants to improve their marks. The higher marks of subjects will be considered in admissions in engineering courses. The application form for the Class improvement exam (CIE) and HSC repeaters will be held in the month of July. The online application for the exam began from first week of June 2020 and will continue until 3rd week of June 2020. The last date to apply for this is 2nd week of June 202 and with late fees payment is 3rd week of June 19, 2020.

Because you are not specific about the university/ college, hence the eligibility criteria of GGSIU Delhi for B.Ed. (Two years of duration) course is given below.Educational qualification: Candidates with at least 50% (fifty percent) marks either in the Bachelors Degree and/or in the Masters Degree in Science / Social Sciences/ Humanities, Bachelors in Engineering or Technology with specialisation in Science and Mathematics with 55 % (fifty five percent) marks or any other qualification equivalent thereto, are eligible for admission to the programme. The University shall follow NCTE guidelines/ regulations for the B.Ed. Programmes for admissions and implementation.
Admission Criteria: Admission shall be on the basis of the merit of the written test /CET. The eligibility criteria of UGC NET for JRF and Assistant Professorship : Candidates who have secured at least 55% marks (without rounding off) in Masters Degree or equivalent examination from universities/institutions recognised by UGC in Humanities (including languages) and Social Science, Computer Science and Applications, Electronic Science etc. (See subjects list Annexure-IV of Information bulletin Year 2019)
https://ugcnet.nta.nic.in/WebInfo/Public/View.aspx?page=5
So, B.Ed is not mandatory for UGC NET. Students in the final year of the Masters programme or waiting for its result can also apply for the exam.
Age: Applicants must not be more than 30 years of age as on June 1, 2020 to apply for Junior Research Fellowship. There is no upper age limit for Assistant Professor. So from above M.Sc. Mathematics students are not eligible for UGC NET but can be eligible for related subjects only (please see subjects list).