Umbreon

If I use Umbreon's "Feint Attack" on an opponent's benched Charizard ex, would the damage be blocked by Charizard ex's "Tera" trait or would it go through since "Feint Attack" ignores all effects on the opposing Pokemon?

Tera traits are considered effects on that Pokemon, so Feint Attack would get through it.

If Shiftry uses "Feint Attack" against Whiscash with "Submerge" Poké-BODY on the bench, which one prevails? The attack or the Poké-BODY?

Feint Attack ignores ANY effect on that Pokémon which attempts to prevent its damage; the attack itself specifically states that the attack's damage isn't affected by Poké-Powers and Poké-Bodies (and other things as well).

If Shiftry uses "Feint Attack" against a Pokémon that successfully used "Agility" the previous turn, would Agility prevent the Feint Attack or not?

No, the Feint Attack ignores any effects on the Defending Pokémon, including Agility, Wonder Guard, etc.

If Shiftry uses "Feint Attack" against a Pokémon that used "Smokescreen" the previous turn, would it have to flip for Smokescreen or not?

Yes it would have to flip. The effect of Smokescreen lies on the Shiftry, and Feint Attack only ignores effects on the DEFENDING Pokémon