Hindi

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

It is a very good degree. You can pursue PG in Hindi literature and post that you can go for Mphil and PhD programme. This will help you get a good start in Govt. college.

For PGT job in schools, you need at least B.Ed qualification. With the only degree, you will not be eligible for any teaching job. Most of the top schools will ask for B.Ed qualification which is the minimum requirement for teaching jobs these days.
Also, if you want government teaching job then you have to qualify TET which is also the for only B.Ed pass candidates.

Yes, you surely can.
A language subject has necessarily to be included in the best of 4, so, you can include Hindi core.

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1.Annamalai University
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The first and basic qualification required for a Ph.D. course is that
you must have completed your Masters degree course.
To apply for Ph.D. you can either directly apply to the professors of
different universities sending them your resume , Statement of purpose
and your research papers. Or you can appear for UGC-NET examination
and then sending your NET scores to the professors under whom you want
to undergo your Ph.D. course.