Can I apply for MBA in this college? I am currently in 3rd year of mechanical engineering but failed in the 1st year.

If you are able to clear your B.Tech mechanical engineering degree with more then 50% marks then you eligible to appear in CAT to seek admission in IIMs. You can appear for CAT in the final year of your mechanical engineering degree.

Yes, you can provided you score 50% or more in your bachelors. But please bear in mind that IIM A is quite stringent in terms of academic requirement while admission. Hence excellent academic record is a pre-requisite for admission into IIM A.