Exam

You can go for the PGCET counselling and if you get the allotment you can get the seat or else you can go for direct admission if you have a minimum of 50% (45% in case of SC/ST candidates) in your graduation with all the subjects cleared.

You must target to score at least 97 percentile to get calls from newer IIMs or top B-Schools like MDI, XLRI, XIMB and S. P. Jain. As you have a good academic profile, you can surely expect few more good B-Schools call.

For Quantitative-
Arun Sharma the complete CAT digest, by Tata McGraw-Hill,
How to prepare for the CAT common admissions test by Muhamed Muneer,
Quantitative Aptitude for CAT and Other MBA entrance exams by Trishna Knowledge Systems (Pearson)
For Verbal-
How to Prepare for the Verbal Ability and Reading

September intake is not possible now so please start preparing for January intake and visa procedure takes almost one month, so plan accordingly.

Admissions in NITs is based on the JEE Mains score.

Yes, there are various colleges in which you can apply for M.Tech without GATE qualification or something like that. I can give you some names like Vivekananda Global University, JECRC University, etc. from Jaipur.

No, if you are doing diploma in mechanical engineering you cannot apply for IIT or NIT because it has stopped taking any lateral entry admission and if you have done your 12th then you can surely apply for the JEE mains advanced.

To get an AIR of 25k you have to get marks between 160-180 and it is also based on reservation.

There are a number of books which you can follow for the GATE, just read thoroughly each chapter whichever book you follow. I am suggesting you some books:-
for production: - Production operations by Allen Robert (vol-1 & vol-2),
for Drilling - Well engineering and construction by H. Rabia,
for Reservo