Guile

The last ability triggers when the Incarnation is put into its owner's graveyard from any zone, not just from on the battlefield.

A spell exiled this way was never actually countered.

If the Incarnation is removed from the graveyard after the ability triggers but before it resolves, it will remain in its new zone when its owner shuffles their library. Similarly, if a replacement effect has the Incarnation move to a different zone instead of being put into the graveyard, the ability won't trigger at all.

Exiling the spell is mandatory. Casting it is not.

Guile's second ability replaces "counter [a certain spell]" with "exile [a certain spell] and you may cast it without paying its mana cost." You have the option to cast it immediately upon its exile. If you choose not to, it remains exiled and you don't get another chance to cast it. If the spell or ability that tried to counter the spell has additional effects, it then continues to resolve.

If a spell or ability you control attempts to counter a spell that can't be countered, it doesn't. Since the spell wouldn't be countered, Guile's ability has no effect on it. The spell will continue to resolve normally.