When would an agent do something different as a result of believing the many worlds theory?
That depends on their utility function.
Sure. The question, there, is whether we should expect there to be any powerful agents with utility functions that care about that.
Would you buy a ticket for a quantum lottery, for immortality?
No. Measure decrease is bad enough to more than outweigh the utility of the winning timelines. I can imagine some very specific variants that are essentially a technology for assigning specialist workloads to different timelines, but I don’t have enough physics to detail it, myself.