B.Ed

Yes you can apply for government job after completing B.Ed through distance and open learning.

The minimum criteria is-
1. Graduate or equivalent Examination with concerned subject as optional subject and Degree or Diploma in Education recognized by National Council for Teacher Education
2. Working Knowledge of Hindi written in Devanagari Script and Knowledge of Rajasthani culture
For more, you may visit then link below-
http://rpsc.rajasthan.gov.in/PDF_Reports_Files/Detail_Teacher_Gr_II_Elementary2013.pdf

Yes, you can pursue B.Ed after completing B.Sc. Graduates in any stream may look to pursue Bachelor of Education [B.Ed] from a recognized institution. Admission is based on either through entrance examination or marks obtained in the qualifying examination, whatever may be the admission criteria of the institution applying for.

Hi,
Yes, you can apply for PGT or TGT with your special degree (I think you are taking about - Integrated Degree).

IGNOU is a well-recognized university for distance education. You can pursue your B.Ed from IGNOU.

Yes, you can enroll for 2 degrees simultaneously.
Refer to the article for better understanding: http://archive.indianexpress.com/news/students-can-now-pursue-two-degree-courses-at-same-time/1155068/

Unless you complete or pass subjects in which you have compartment you will not be eligible to seek admission in higher education whose eligibility criteria is graduation degree.

Please clarify your question. What do you mean by Bombay Art?