IIM Ahmedabad - Indian Institute of Management

You can opt for one specialization at a time. However you can choose to major in another area as well. Also there are people who do multiple MBAs to specialize in multiple fields.

It would totally depend on you as these big businesses hire from almost all college. Also if you want to start your own business, then you should do whatever you are comfortable with.

No in cases of integrated degree, you receive two degrees. However in each of the degrees it is mentioned that this degree has been awarded under integrated degree program.

Yes. Maths is very useful for CAT. One of the two papers in CAT checks your quantitative abilities. However majority of questions are from class 9th to 12th maths. Also you should not base your life decisions on requirements of one examination. If you think your time and energy are useful in other subjects than maths, I think you should take that up.

IIM Lucknow is behind IIM Ahmedabad, Calcutta and Bangalore in rankings.

It would totally depend on you. First of all you should identify your strong areas and your weak areas. This would only be done if you have evaluated your mock tests properly. Remember that the key to crack CAT is SPEED and ACCURACY. Hence it is important that you solve as many questions as possible correctly. And this would only come with practice. Hence in short, practice as much as you can.

CAT exams are more about practice. It is more speed and accuracy than knowledge (unlike JEE). Questions are essentially very easy and if given sufficient time, you will be able to solve all the questions yourself. Hence you should focus more on practice under exam like situations. Post mock tests you should always evaluate what went right and what went wrong. Try to develop an understanding of your strong areas and the weak ones. Improve on weak areas and continue to do well on your strong areas.