B.E. Eligibility

No, you cannot get admission in Sathyabama just with your class 12th and class 10th mark. You will have to write the Satyabhama entrance exam to get admission here at Satyabhama University .
You can apply for the entrance exam using our college website: http://sathyabama.ac.in/engineering-entrance-exam
This is the official website of our college. You can also get the other details like college fees etc. in this website.

Eligibility Criteria for Lateral Entry:
Candidate must have passed Diploma in Engineering/Technology from a recognized board.
https://www.shiksha.com/b-tech/course/b-tech-lateral-entry-mepco-schlenk-engineering-college-tamil-nadu-other-243401

I'll tell you the whole admission procedure in detail.
So, first of all, you'll have to register yourself on cu FastTrack and one unique I'd will be generated to you.
After that you'll have to register yourself for the admission test and if you clear that test then you're eligible for the admission and you can also avail scholarship from the University that will be compensated to your academic fee only.

Use Jee Mains College Predictor to know the same:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor

Plese tell me first which college varun

Please visit this site first
https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor&ved=2ahUKEwjd0NzModzhAhUZfSsKHfLtA6UQFjAAegQIBBAB&usg=AOvVaw1N9BzFtyUCP4JU-2UQHQ0H

Yes, you are eligible. Eligibility for BE (Lateral Entry ) -
Candidate must have passed Diploma in Engineering/ Technology or B.Sc. From a recognised university.

Hello, Yes, it is. Here are the eligibility criteria for JEE Advanced AAT Exam:
1. Only those candidates who have qualified in JEE Mains JEE Advanced 2019 are eligible to appear in AAT 2019.
2. Only Candidates who opted for JEE AAT during the online registration process. Click here http://bit.ly/2UIs2oT to check the complete details of JEE Advanced AAT Exam. Best of Luck!