XLRI Xavier School of Management

Admissions in XLRI are done through XAT. You need to secure at least 95 percentile in XAT to be considered and called for an interview.
CAT is for admissions in IIMs and numerous other colleges across India. Some top colleges admitting via CAT are SP Jain, MDI Gurgaon, IMT Ghaziabad etc.

Admission to XLRI programs in the field of management is through XLRI's entrance test known as Xavier Aptitude Test (XAT). XAT score is also used by multiple other management colleges for admission to their MBA/ PGDM.

There is no quota for students from any college, including Xavier's.

Yes, in most of the cases separate registration is required. Please visit websites of all the colleges you want to apply through XAT and find the application procedure for each.