CAT

It will be difficult to get into a government institute for MBA at this score. If you have a good academic and work profile you have slight chances of getting into IITs (like IIT-Kanpur, IIT-Madras).

Hi,
You can apply in Jaipuria-Lucknow, Bengal Institute Of Business Studies, ITM, Doon Business School, NDIM-Delhi, Amrita Business School, PSG-Coimbatore or Prestige Institute-Indore.
Check the below link-
https://www.shiksha.com/colleges-accepting-cat-scores-range-60-to-70-percentile-article-11984-1
Also, check this link-
https://www.shiksha.com/mba/resources/application-forms

You have slight chances of getting calls from IIT-Kanpur / Madras, if you have applied for them. It's difficult to get calls from IIMs. Just check for any mails you receive from newer IIMs like IIM -Srinagar, Bodhgaya and Sambalpur. Apart from these you can apply for colleges like IFMR, Great Lakes, ITM-Mumbai, Christ University-Bangalore and others.

For the year 2016-17, cut off was about 89 percentile.

Yes, you are eligible as per DTE norms.

No, 50% is the minimum eligibility.

Every reputed institute follows the entrance exam, it is basically to pick the cream crowd from the lot.
There are institutes that take direct admission too.

Marks are not considered while applying for CAT, but institutes check marks of 10th and 12th before the selection of the candidate.

You can apply to IBS Group, IMT, UBS, Welingkar, SIES, NL Dalmia, IBS, Indira and Balaji are some of the institutes for the same.

CAT exam has already been held for this year, the results have been declared.