IIM

Yes. In my knowledge, IIMs consider only full time, regular bachelor's program and not the distance programs.

There is no available data for this. However TIME and IMS looks like the one with highest number of selections.

Yes there are. They provide financial assistance to families with income less than a specific amount. Generally 3-5 lakhs.

Minimum salary - 9 lakhs per annum

I don't think you would. You would not clear BE cutoff, which is usually 60%. But these criteria change every year and if you do really well in CAT, you might get calls from few IIMs that don't give much weightage to BE marks.

If you have cleared all the backlogs and have received the degree in time, you are eligible to appear CAT given you fulfill minimum percentage requirement. Here is the details about cutoff requirements.
http://www.iimahd.ernet.in/users/admission/files/Admission Procedure_2016_2018_August_10.pdf

I don't think you stand a chance of getting into top six IIMs with 85 percentile. For general category students the cutoff goes as high as 98 percentile. However if you from non-engineering background, the cutoff are slightly lower. I would suggest that you give your best shot at CAT and not worry about cutoff now. Higher you score, better chances you have of qualifying.

your ajswer would be yes. It has been the most preferred and prestigious combination which makes a strong manager with technical and management skills.