1. Why were the trips to the end of the world different for everyone who went?
The many-worlds theory, I'm guessing. Every outcome with more than one possible state results in multiple different universes, one for each possibility. Enough variations create wildly different realities. Given enough time from a specific starting point and things could end up almost anywhere.
2. Why was the world in a seemingly perpetually accelerating dervish of disasters and plagues?
A cynical view of human nature and the second law of thermodynamics. Things fall apart, the center cannot hold, etc.
3. Were the time-tourists changing the future by visiting it?
Possibly, though likely not much. Might be that visiting the future changed the time travelers, which might then change the future(s) available to later travelers.
4. Were the repeated trips to the future causing the natural disasters/plagues/etc to increase as more people visited the end of the world, or was that just the world they lived in?
I'm guessing it's just that entropy always wins, personally. The story doesn't really address the idea, per se. It's more a story about a human reaction to tragedy, i.e. it's the end of the world, but no one cares because they're busy watching... the end of the world.