I have a few questions (as the title implies):

1. If I use the battlefield Mean Streets to decrease In the Crosshairs' cost by 1, could I then also use Calling in Favors to decrease its cost down by another 1 thus only paying one resource, or would I not be able to use Calling In Favors since I already decreased its cost down to 2?

2. If I play Art of War and reveal my opponent's hand to find a card with a cost of 3, and he does not have a die showing a value of 3 in his pool but I do, since it doesn't say "you may remove a die," do I have to remove my own die?

3. In the case of a card like Deflecting Slash, which reads "Activate one of your Blue characters to remove a die showing ranged or indirect damage," do I also get to roll that character's character and upgrade dice into my pool, or am I using its activation to remove that die?

1) No, for the reason you provided.

2) No, for the reason you provided.

3) You are still activating the character. Resolve all effects that trigger from them activating and roll their character and attached upgrade dice into the pool. Deflecting Slash would read "exhaust one of your blue characters" if such were the case.