MAT

Without CAT/MAT scores you will not be admitted to the MBA programme. Kindly attempt these tests and then apply.
All the best.

Do you have a CAT/XAT score also? If you have a decent CAT/XAT score also, I am positive about your admission to ASB. I am afraid the admissions for the current year has been closed. Please contact the admissions department right away.

I assume you mean to say that you got 58.8 percentile in MAT. In order to get into XLRI, you need to give XAT. Upon receiving the call from XLRI, you will have to clear the GD and PI rounds. The cumulative scores of XAT and GD-PI will be taken into account for the admission in XLRI.

Yes, you are eligible to apply.

No, KJ SIMSR doesn't consider MAT score for admissions.

Hello! If you want to pursue MBA full-time then you need to appear for at least one of the several entrance exams whose scores are considered by the colleges for their respective admission processes. There is no other way one can get admission in a reputed college apart from it.
https://www.shiksha.com/mba/articles/20-mba-entrance-exams-you-can-take-this-year-blogId-11473
Kindly visit this link to know about the various MBA exams conducted in India every year.

Below are some suggestions. The best book for the preparation of MAT exam.
1. English/verbal ability S. P. Bakashi
2. Mathematics - R. S. Agarwal
Good luck for exam and your efforts. You can also follow the books of Arihant Publications.