CAT

See there are a few State Government Colleges in Uttar Pradesh (UP) but you can try in:
Harcourt Butler Technical Institute (HBTI - Kanpur), Institute of Engineering and Technology, Lucknow, Dr APJ Abdul Kalam Technical University, Lucknow
Bundelkhand University, Jhansi
Bundelkhand Institute of Engineering and Technology, Jhansi
Mahatma Jyotiba Phule Rohilkhand University, Bareilly
Veer Bahadur Singh Purvanchal University, Jaunpur
Institute of Cooperative and Corporate Management Research and Training, Lucknow
Indira Gandhi Institute of Co Operative Management, Lucknow
SPCJ Institute of Commerce Business Management and Economics, Agra.

No.
I am sorry but you have to have to go through CAT or XAT.

The CUSAT CAT result will be available in this month itself in the official website of Cochin University of Science and Technology (CUSAT), www.cusat.nic.in. The CUSAT CAT Result 2018 will contain your score/marks in the test and thus according to it, you will be placed in the CUSAT CAT rank List 2018.

Some important CAT preparation tips are mentioned below:
1) Build your Basics
2) Solve Practice questions & Sectional Mocks
3) Overcome the weaknesses
4) Find out more than one way to solve questions
5) Devote equal time for preparation on all 3 CAT sections
CAT 2018: Simple tips to optimize exam preparation: https://www.shiksha.com/mba/articles/cat-2016-simple-tips-to-optimize-exam-preparation-blogId-13034.

Three things to keep in mind in group discussion
1. Always try to initiate the discussion with such a point so that the other person must continue further from you point where you ended
2. Always try to tackle opponents points with your point is a relevant manner
3. Make the whole panel to agree on the same point said by you.

It depends on when you wish to appear for the exam. You need to start with 1 or 1.5 years prior to the year in which you want to appear for the exam.

You need to start preparing at least one and a half year prior to when you wish to appear for the exams. So, if you start preparing halfway down the 2nd year then at the end of the 3rd year you are good to go.

No, it's not necessary to apply for CAT to get admission for 5 years integrated course in CUSAT, you can also get admission through CMAT and MAT scores.
Details are mentioned below:
https://www.shiksha.com/mba/exams/cusat-mba