Auto-switching between power sources with multiple voltage outputs
Looks in manufacturers catalog for oring controllers and battery backup switches (every brand has its cute name like TI eFuse, Linear PowerPath and so on). These are way better than the huge diodes you would need (some part even interfaces with the battery charger to help it).
As for the output side I agree with your plan. A main 50V rail for the power amp and the other rails from which step down the other branches is a good solution.
The 120VAC mixer is an exception, it makes no sense to step down to 50V and then invert it to 120VAC (a can of worms in itself), so I'd power it directly from the oring controller output.