By the rules, there isn't ever a timing issue between Block and Falcon Precision. Block is revealed during the Avoid Attack Step due to it's mandatory reveal requirement. It can not be revealed at any other time without being canceled and discarded. Meanwhile, Falcon Precision as a voluntary reveal can only be revealed after a phase, step, or activation. Which means it can not be revealed DURING the Avoid Attack step. It would need to be revealed, at the very latest, after the Roll to Miss step. So, if you go through everything step by step sequentially, there is never a timing issue and the Falcon Precision would always be revealed first (before you know there is a Block).
Now, practically, when you're at the table or on OCTGN, you don't typically go through every single step. Perhaps you should at a tournament, but you're already pressed for time. So, I can't answer how a tournament would handle the situation, for that I'd recommend getting Silverclawgrizzly to weigh in, but personally if the person had revealed the Block prematurely I'd let them keep their mana because there's no way they could reveal it before the Falcon Precision without it fizzling. Meaning they clearly thought they were already in the Avoid Attack step.
With Daze and Fumble, the rulebook states that when an attack Misses, it skips to the Additional strikes step. Block would therefore not be triggered because it happens during the Avoid Attack step. Dodge is a little more problematic though. Because it's an Academy card, and Academy has a different series of steps than Arena, it is written in such a way that says it actually gets revealed BEFORE Daze even occurs, immediately after the Declare Attack step. This means that it would get used up despite a Daze or Fumble. Because it triggers before Daze happens, and it triggers at the same time as Fumble and the trigger is mandatory even if Fumble gets revealed first and makes it irrelevant. But that is how it would work according to the rules.
Edit: I deleted an opinion statement that was maybe a bit too harsh. I just really dislike that Academy and Arena are such very different games but they keep shoehorning them together. I really think they should be completely separate.