MAT

MAT has only a few good colleges some of them include Christ University-Bangalore (cutoff 600+ marks), ITM-Mumbai (above 75 percentile) and Balaji Institutes-Pune (cutoff above 70 percentile).

Below mentioned are the best management colleges accepting MAT score:
1. Christ University – Bangalore.
2. XIME – Bangalore.
3. XIMR – Mumbai.
4. BIMTECH – Noida.
5. Flame University – Pune.
6. Balaji Institutes - Pune.

The exact date for the interview will be mentioned on your admit card which will be released around 25th. Usually the the candidates are divided into 2 groups and the interviews are scheduled on consecutive days for each group(i.e, within the dates mentioned on the website). So, you can either wait until you get the admit card and see when your interview is scheduled or maybe call the admission office and request them not to schedule it on 29th. I can't guarantee a positive response but at least you can give it a try.

You can consider:
1. IMT – Nagpur.
2. IMT - Hyderabad.
3. Christ University – Bangalore.
4. BIMTECH – Noida.
5. Balaji Institutes - Pune.

No, KJ Somaiya accepts CAT, XAT, CMAT and MH CET scores for MMS and PGDM programs.

Yes, you will get a good job after completing MBA from a decent college. You can appear for other national level exams like CMAT, ATMA, etc.

You can look for other colleges too, which you can get at this score like:
1. ITM – Mumbai.
2. XIME – Bangalore.
3. Welingkar.

Yes, you are eligible and have Decent chances at PSG as its cut-off is above 550 and near 620. Overall you have fair chances, so keep hopes high and wish you good luck.

Apeejay is not recommended at all, neither should you take a drop and reappear. I am sending you a list of colleges at your MAT score below. Kindly check their cutoffs and decide accordingly.
http://www.aimamat.in/mat-college-list-percentile-wise/

You can opt for Christ - Bangalore and PUMBA – Pune.