Chemical Engineering

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Hello Chinmay,
Among all the colleges, you have listed above Information technology at IIIT Allahabad is much more better option than any other, because IIIT Allahabad is very much known for its excellent teaching and superb placements records.
Best wishes.

Yes, you can definitely appear for PSC exam Kerala after studying B.tech in Chemical Engineering only you have to some fees for exam. Best of luck for your future and if any Query feel free to ask.

Yes, you can definitely work in Kerala by taking the Degree of Chemical Engineering. The university not matters while doing the job but you need experience for doing job. Best of luck for your future and if there's any query, feel free to ask.

Fees for Shivajirao S. Jondhale College of Engineering (SSJCOE) for B.E. in Chemical Engineering is Rs 86,000 per year

Yes, Of, course ! You can attend the GATE which is an entrance exams for getting admission in IITs for the further post graduation. Graduate aptitude test of engg. is an entrance exam held by IITs to qualify the students to study there.
Hope this helps you.

Technical University of Munich
Technical university of Berlin
Karlsruhe Institute of Technology
RWTH Aachen University
TU Dortmund University Technische Universitat Darmstadt
Universitat Stuttgart
FAU Erlangen-Nurnberg
Abert-Ludwigs-Universitaet Freiburg
Hamburg University of Technology
These are the universities for chemical engineering in Germany.

It all depends on your liking towards any of the branches that you have mentioned. NIT Trichy is one of the best and the oldest NIT amongst the other NITs. It can definitely fetch you good placements. Also, CSE at NIT Trichy is the best.