Pune

If you're looking for roles into accounting and financial management, then its a good choice.
Overall placements assistance is average and good companies do visit.
https://www.shiksha.com/getListingDetail/21643/institute/college-Indian-Institute-Of-Cost-And-Management-Studies-Research-Indsearch-Pune-India

It is not necessary to provide all the specialisations. A college keep in mind, the demand, number of students, faculty, infrastructure and availability.

A person usually pursue MBA to have better career options. A specialisation has to be picked keeping in mind ones interest area and inherent personality traits. If a person does MBA in any other specialisation then he may not enjoy the profession that he picks up as a career. Many a time this leads to a situation wherein the person reviews the career options at a later date. My advice to you is that keeping in mind your interest areas and personality type pursue a specialization.

Symboisis and Pumba are like cream of Pune. Rest all are average.

MAAC has better placements, faculty and result oriented studies.

COEP and VIT Pune are the best lateral entry colleges which you can opt for or you can give LEET examination to get more colleges.

You can opt for either Bull's Eye or TIME. Both the institutes are good for the preparation of CAT exam.