Shooting an arrow at a target during the competition in a tourney. There are plenty of examples.
Once again, the rules say what you can do "during a normal round;" they (rightly) are not so restrictive outside combat.
Given that your insistence on inventing non-existent restrictions on out-of-combat action results in an absurdity -- namely, being unable to cast spells out of combat -- why do you insist on doing it? It's quite odd.
Let's see - I didn't insist that you are not flat footed outside of combat. When the text states that you are until your first turn in the initiative order. I did come up with how to apply existing rules (and justification) for how to apply feather fall when you fall due to a trap outside of combat - or before your turn in the initiative order.
Now the statement that the text of feather fall allows you to use even when it is not your turn is also within the existing rule structure in that specific text overrules general one. So even after the spell errata to make feather fall an immediate action that specific spell description text still can apply to supecede the immediate action restriction, again within the existing rules structure.
As far as tourneys go - do you not use a round structure to cover those? Isn't there initiative involved?
Combat system structure is supposed to apply to all types of encounters not just those involving combat - so IMO your single example of making an attack roll outside of combat misses the mark since the example is still within a combat round structure.
Now the DMG on page 25 does talk about combat actions outside of combat. But it does state that "As a general rule, combat actions should only be performed in combat - when you are keeping track of rounds and the players are acting in initiative order."
Basically this says that the combat structure does not only apply to strictly combat situations but to anytime where you are following initiative order and keeping rounds.
Now lets try looking at how to handle things outside of combat differently.
Assuming (not that I necessarily agree with this assumption) that flat-footed applies only during the initial round of combat. You should still not be allowed to have your Dex bonus to AC since per (PHB pg 8) "Armor Class (AC), provided that the character can react to the attack"
PHB pg 134
"An attack roll represents your attempt to strike your opponent on your turn
in a round."
So if attacked by a trap your AC should depend on whether or not you are aware of the trap or not. If you are not aware then you should not be able to apply your Dex Bonus (consider it denied, so that Dodge bonuses do not apply either).
The DM can start an encounter whenever he wishes to. Excuse me, the DM determines when an encounter begins (DMG pg 22).
All of my statemets are base on RAW and applying the RAW to similar situations in a consistent manner (as allowed by and encouraged by the DMG)