1) The challenge about throw weapons requires you to use missiles while dueling. I am not sure if studio fights count. Every missile used up counts towards the challenge.
2) Yes, that is my suggestion. Courage trophies cost too much, energy trophies cost too little. The members of the top 10 studios will tell you that they fight like hell to take energy trophies, will take courage trophies if nothing else shows. It is worse for the studios (like mine) which can typically hold 4 or 5 trophies if they want to. We fight like hell to avoid taking courage trophies, because they take up a slot that should hold an energy trophy.
Another studio boss I regularly converse with called courage trophies a communicable disease. Used to be, once you got one, it was damn hard to get rid of it. But now, using 400 diamonds to get rid of a courage trophy is well worth it. It's like penicillin entered the game.
Why is energy so much better? Energy lets you directly make your avatar better. Courage let's you duel more... big whoop. Dueling is mostly a waste of time, the rewards are barely worth the time put into them, and glory is over-rated. (Glory does not directly measure combat effectiveness - it mostly measures your desire to gain glory.)
A studio that gets powerful enough to hold 5 energy trophies has gotten to an unbeatable state. As long as the members of that studio stay active, no other studio can catch up. That is why many servers have two dominant studios, that can hold full boats of energy trophies - and everyone else trails behind. (There are only enough energy trophies to let two studios fill up completely.)
My proposal would not completely solve the problem of snowballing win conditions... but it will give more studios a fighting chance if it is harder to actually hold five energy trophies.