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Regarding the firm it will not be possible to make comments, I have heard about Amazon and Delloite as they come to colleges for campus placements. The choice would depend on where you wish to see yourself a few years later, your area of confidence, the pay and perk they offer and the working environment offered.

Hardware and Networking Courses may be of interest to you. You may also pursue some international certifications such as CCNA, MCSE etc to further your prospects.

IIMs do not consider this point for admissions.
However, if we look at top IIMs, you'll see that more than 60-70% students will be from top colleges.
Ideally, if you score good in CAT and perform well in interviews, not being from a reputed college, will not create any problems.

You must clear your backlogs before you can pursue higher education. CA/CWA etc can be pursued along with graduation if you have plans to offer consultancy services.

No there is no other way for Indian to get admission in nit calicut. Jee main is only option to get admission through counseling during July every year.
PS: if you are a DASA candidate ( outside India) there is 5% seat are reserve for them and they are coming without giving jee main exam.

A Score above 330+ can get you into Excellent universities. 315 - 330 will get admission from above average Universities. 300-315 Score for Average Universities.
Go to the link below to check which all colleges you can apply with your current GRE score:
http://www.msinus.com/content/gre-universities-486/#.Vl19CFK3iAk