Compressor Temp/Press Impact - Loss of Cooling to All Interstage Coolers

1. Is there a theoretical equation to calculate the theoretical maximum temperature at each stages discharge, and check against the MAWT or design temperature of casing / suction KO drums?

This has all the earmarks of a Home Work question. Did you take notes during the course? If not, you may want to start here. Good Luck

"trying to analyze as part of hazard review"

1. In such centrifugal compressors there are two vital protections, (1) Cooling water flow low/ too low and (2) Gas cooler outlet temperature high/ too high. If you have these protections installed, no need of any calculation required because compressor will trip before.

As alarm and trip set points are fixed, it is not required to account the spill-back valve is recycling back hotter gas back to the suction of each stages when determining the max temperature at the discharge. Is it each stage or only final stage?

2. Not required, as normally designer takes care for it up to trip set point temperature.

"suction temperature has little impact on the discharge pressure since the power is constant." How power is constant? Suction temperature can influence flow rate, discharge pressure as well as power at similar conditions.

No any feedback from OP.