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For example, a wizard casts scrying to peer into the big bad's lair, where his pet umber hulk happens to be lounging. Would you rule that the caster must deal with the effects of the umber hulk's gaze attack since his vantage point is within 30', or would he be safe from the umber hulk's confusion ability since he physically is still miles away?

I have a feeling it's the latter, taking the rules strictly as written, but I'm open to the possibility there was a clarification published somewhere that I'm not aware of.

I'm also curious if the answer would be any different for sound-based effects, such as a chasme's droning or a satyr's pipes.

RAW, no.

Umberhulk's Confusing Gaze specifies that it takes effect when "a creature starts its turn within 30 feet of the umber hulk and is able to see the umber hulk's eyes." Scrying an umber hulk would satisfy being able to see its eyes, but the wizard is well outside of 30 feet, and isn't starting their turn aside from that. A chasme has the same qualification. I can't find relevant information for satyrs online, but I'll crack open my monster manual when I get home. Gazes aren't recognized as separate types of attacks in fifth edition, so a blanket judgement can't be made.

However, this feels very silly to me, personally, and I feel this is a situation where the rules should bend to in-universe logic. In the case of the umber hulk, I wouldn't hit the wizard with confusing gaze unless the creature had see invisible and noticed them, but I would target them with a chasme's drone.