Now upon exiling a creature with a coin counter from Athreos, Shroud-Veiled with Flicker of Fate, does the triggered ability of Athreos, Shroud-Veiled take over and and stop the rest of the Flicker of Fate's effect?
No, after a triggered ability triggers, it is placed on the stack the next time a player would gain priority.
603.3. Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority. See rule 117, “Timing and Priority.” The ability becomes the topmost object on the stack. It has the text of the ability that created it, and no other characteristics. It remains on the stack until it’s countered, it resolves, a rule causes it to be removed from the stack, or an effect moves it elsewhere.
No. Spells and abilities in Magic aren't interrupted like that. The specifics on triggered abilities going onto the stack are covered by rule 603.3 in the Comprehensive Rules, triggers see what happen during a spell resolving, but wait until someone would get priority to be put on the stack in Active Player (the player who's turn it is) /Non-Active Player (every other player, in turn order) order. This is how the situation would resolve:
- A player casts Flicker of Fate targeting a creature with a coin counter.
- All players pass priority and Flicker of Fate starts resolving.
- Flicker of Fate moves the creature into exile.
- Flicker of Fate returns the creature to the battlefield.
- Active player would receive priority, state based actions are checked and once all of those (if any) are resolved, Athreos' trigger is put on the stack.
- Players all pass priority and Athreos' trigger starts to resolve.
- Athreos' triggered ability tries to return the creature from exile, but the creature is not in exile, it does nothing.
Once a spell or ability starts to resolve, nothing happens until the entire ability finishes resolving, Not even if say the first half of the ability would cause someone to lose the game, the spell finishes resolving and then whatever that spell resolving triggered happens.
As a side note - If Athreos' ability had been written "If a creature with a coin counter would die or be put into exile, remove all damage and coin counters from it and put it on the battlefield under your control instead" then this wouldn't be a triggered ability anymore, it would create a replacement effect and Athreos would hijack the Flicker of Fate in that case, moving the creature to whoever controls Athreos' battle field rather than flickering it to exile and back. (The damage removal is needed because it never leaves the battlefield, and without it it'd likely just die again)