M.Sc Botany

National Eligibility Test, which is conducted by the University Grants Commission for the lectureship jobs in central government colleges, does not have Botany or Zoology as one of the subjects. This, I believe has lead to confusion for you regarding the subject. The answer to your questions is pretty simple. You will have to choose 'Life Science' as the subject for the CSIR UGC NET entrance exam. Regards.

Yes, after getting your graduate degree, you can easily take admission in M.Sc chemistry or botany.

Please refer to the link provided below for details regarding M.Sc course in Wisdom School of Management. You should be able to download their brochure which contains details of availability of courses from this institute.
https://www.shiksha.com/science/course/m-sc-wisdom-school-of-management-lucknow-120692

Please refer to the link provided below for details regarding admissions in M.sc. Botany in Rajasthan University. Additionally, you can also download their brochure which contains important information regarding admissions, course eligibility, entrance exams (if any) etc.
https://www.shiksha.com/science/course/m-sc-botany-university-of-rajasthan-jaipur-31237

Yes, you are eligible to apply. You should possess MA/ M.Sc/ M.Com in the relevant subject with B.Ed.

The skill assessment test is to assess your written and communication skills. It will consist of a test on written skills and communication skills (generally a common topic about current happenings) and logical reasoning.