MAT

Yes you are eligible to apply IES for PGDM programme. IES considers composite score across various parameters, aptitude score is one of the parameter.

Best colleges in Hyderabad accepting MAT is Osmania University. Either from this, no one is better but one more is IFCAI Business School, but they are not accepting it.

Go through the link given below:
https://www.shiksha.com/mba/ranking/top-mba-colleges-in-india-accepting-mat/2-2-0-0-306

Yes, you need to have CAT, MAT, GMAT or XAT scores or have to appear for the entrance examination of the institute but that totally depends on the institute/college in which you wish to apply. Do check the eligibility criteria before applying.

Yes, MAT score is valid for 1 year.

There are a lot of colleges with such approach. For instance, to get admission in Narsee Monjee, you need to clear NMAT. Likewise, to get admission in Symbiosis, one should appear for and clear SNAP.

Hi! There is no pre-defined syllabus for the exam. However, based on past trends, questions on mensuration are asked in the exam.

You can consider following colleges:
Ramaiah Institute of Technology, Bangalore.Indian Institute of Management Bangalore.
BMS College of Engineering, Bangalore.
PES University, Bangalore.
Christ University, Bangalore.
Bangalore Institute of Technology, Bangalore.Dayananda Sagar College of Engineering, Bangalore.

This can't be predicted because in my opinion, the composite score also depends on the number of applicants that particular year.

Although MAT is a good option, I would suggest you to prepare for other entrances as well so as to open more options. Hope this helps, all the best!