GATE

Revise basic concepts and try solving previous years GATE question papers. You can try solving study materials of Made Easy.

1.) Arihant publications.
2.) GKP.
3.) ACE academy books.
4.) Elite academic books.

Yes, you can take GATE exam with mechanical. The GATE authority allows two paper combination for various papers in case any candidate wish to appear in more than one paper in the same year. The list of two paper combinations for GATE can be checked on the Shiksha page as well as the official website of GATE. 

Yes, it is possible. PSUs generally recruits on the basis of GATE scores.

Kindly refer to the link mentioned below:
https://www.shiksha.com/gate-exampage

Fellow Program in Management (FPM) is offered by IIMs in which students are selected on the basis of GATE score.

Refer link below for details:
http://www.inspirenignite.com/non-gate-or-without-gate-m-tech-admissions-2016

Kindly check out this link-
http://gateguide.in/gate-previous-year-papers-solutions-free-pdf-branches/

Yes, you can. Try to revise the basic concepts of subjects which are in your course of GATE. Solve as much problems as possible on those concepts. You can try problems from Made Easy materials or Career Launcher study material. Also, solve problems from last 10 years GATE papers.