Is loot quality affected by difficulty level (or challenges/strategy mode, etc…)? Are there better rewards based on any of these things? Is beating a level on hard going to give better items on average than on easy?
Here are the things that I know for sure affect loot drop quality
Here is what I think affects it (can't confirm for sure)
The boss rewards do have a higher chance of dropping a better quality item on higher difficulties.
As for challenge maps, most challenges give different rewards based on difficulty level but those rewards are unaffected by character level.
In addition to what yx mentioned, loot also seems to get better the map wave number is. A hard survival game in wave 1 will drop worse loot than that same game on wave 15.