If a Raichu (or whatever) does Agility and flips heads, does an attack that would do damage to benched pokemon, or some other effect that does NOT do damage or something to Raichu work?

Yes it would, while all attacks are considered against the defending Pokemon (which is why something that prevents attacks would stop it in this case), Agility only prevents the effects of that attack if they affect Raichu only.

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