Computer Science and Engineering

I don't know who told you about Graphic Era but Graphic Era is not a good university because I am from Dehradun only I would suggest you to take admission in YMCA because a friend of mine is studying there and it is a very good college. Instead of taking admission in Graphic Era you should go to YMC

Yes, of course you can get a seat in Alagappa Chettiar Government College.

The minimum required cutoff to join is 68%.

Refer the following link:
https://shiksha.com/b-tech/resources/jee-mains-college-predictor

As per my views, history will be better as compared to political science because history can be interesting and political science can get boring sometimes. However, as you said you are in arts stream I doubt you will not be eligible for CSE engineering because PCM is a must condition for the eligibi

The fees is 117k for 3 years.

Firstly, for cut-off, there is no need of percentage in 12th and according to your percentile, there will be 30% chances of getting CSE but you can go for CCE, which contains more than 50% syllabus of CSE and has good placement records.

If you want to pursue M.Tech in IIT Bombay then it does not depend, which was your earlier college you did B.Tech from. You can do both the degree programmes from different colleges.

Yes, it accepts JEE Mains rank, but it's not possible for you to get admission in CSE in that college. To know in which other colleges you will get admission use the predictor tool:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor