GATE Eligibility

Eligibity criteria for appearing on GATE is that candidate must have completed Bachelors degree in Engineering/ Technology (4 years after 10+2 or 3 years after B.Sc./Diploma in Engineering/ Technology) and those who are in the final year of such programs. Even if you had backs, you are eligible to appear for GATE.

If you want to become a scientist at ISRO, you should appear for the ISRO qualifying exam rather than appear for GATE.
The eligibility:
BE/B.Tech or equivalent qualification in first class with an aggregate minimum of 65% marks or CGPA 6.84/10 (average of all semesters for which results are available). Minimum of 65% marks or CGPA 6.84 in section B alone for candidates with AMIE/Grad IETE qualification.
The qualification prescribed and the benchmark are only the minimum requirement and fulfilling the same does not automatically make candidates eligible for written test.
To answer your question, GATE syllabus for both would be the same.

Yes, you're eligible for GATE. As long as your age is below 25, you also get to apply for the various PSU's via GATE. If not, then you can use the GATE score elsewhere to your advantage.

Yes, after your B.Tech, you are eligible for GATE exam as this exam is for higher study in IIT and NIT. So, you have to study well because the level of questions is high in GATE.
All the best.

Yes, computer science engineering is eligible for GATE entrance exam.

Yes, you can. IITs don't have any interest in the college you opted for UG. They only consider your GATE score.