B.Tech

Manipal university, Jaipur is a good university for engineering. But, many others institute are very good in Rajasthan like LNMIIT, Anand ICE, Jaipur according to placement. From my opinion Anand ICE, Jaipur provides world class infrastructure, best higher educated faculties & better placements. It's also provide abroad internship to students during the B.Tech. So, my suggestion is that you should visit one time at Anand ICE, Jaipur.

You can study distance B.Tech in mechanical engineering from SSK College in Chennai.

Yes, it is not tough to get in there. Apply for it and all the best.

Assuming you are from General category, getting into CS is tough for you as for out state students, last year's cutoff was 180 and that too for home state. I would suggest you to get a college in Jaipur or Udaipur with home state as most of the seats in sits are filled by home state itself. Else, try for private colleges with better placements.

You have to apply through TNEA 2018 for getting Admission in Anna University. There is a different quota for Non-Tamil Nadu State Students. More details about TNEA 2018 is on the link below.
https://www.shiksha.com/university/anna-university/exams/tnea
https://tnea.ac.in/adata/index.php
Apply for TNEA and sit for Counselling. More details are on Anna University's Website.
https://www.annauniv.edu/
You can also call or email them for more details. Number below is contact number of Anna University.
044 2235 1126
All the Best.!

SGSITS is good for placements. Being an autonomous institute, the companies prefer it over, private colleges of the region, The highest package one recruited was Microsoft through campus placements of around 24 Lacs per annum. Moreover, if you are in one of the core branches of GSITS, you'll get recruited as the number of students are not more than a certain number and every year, there are around 95% placements. Best of luck.

Both have their own career prospects. The old college has a very good experience but lack of new technologies while the new college has lack of experiences but having new technologies. You can choose anyone because both are beneficial for Computer Science Engineering.

Yes, you have chances to get in.
All the best.

You can get government college. But first choose your destination course which you are want to do. Please visit the link given below for best colleges based upon your JEE rank
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor?utm_source=connect&utm_medium=CR_JEECP&utm_campaign=6919840.