Anastasia "Dee" Dualla Once-Per-Game Ability Timing
The official sequence of events for resolving a Skill Check:


Read Card
Play Two Cards from Destiny Deck
Play Skills
Shuffle and Divide Cards
Total Strength
Determine Result


I've generally interpreted the "before making a skill check" phrase to mean the time between steps 1 and 3 (when the Destiny Deck cards come out usually isn't important). So things like Investigative Committee, Reckless and Dee's ability should ideally be used during this time. However, things sometimes happen a little fast and people start throwing in cards before someone has a chance to announce an action. We're usually pretty lenient and let people do things after step 3 starts if it doesn't seem like they're gaming. 

I don't think there's any specific sequence between Investigative Committee, Reckless, Dee's ability, and the like. I think they could happen in any order. Generally we play it kind of loose and the first person to act goes first; normally it doesn't matter. Ideally, everyone that has something to do should indicate that they do. According to the rules, it's up to the currently player to decide the order.