I'm always annoyed when akali, eve, twitch, or shaco goes stealth as an escape mechanism. Got me thinking...

If Shen were to dash through them, does that initialize the taunt -> force auto-attack -> and cause them to reveal?

I know they'll still get hit by skill shots and AoE abilities - and what about something like Rammus's powerball - would he hit or just roll through them? It it hits....could be an effective way to track someone down.

Yes

For Rammus' powerball, you'll just hit them and stop for apparently no reason. So, Amumu's bandage toss will likely pull you to an invisible Eve as well, even though you can't see her.

Shen's dash will taunt them out of Stealth.

Yes, all AoE or skill shot abilities will affect stealth units.

One thing to keep in mind, is the taunt itself doesn't remove stealth, so if you're taunting a stealthy, but you ran also dashed through a wall/away from them and stayed out of range, all they'll do is run towards you while still stealthed; they need to actually complete an auto attack to have their stealth removed.

Yes!

Small thing you might want to note, Shaco's invis is a blink (short teleport) as well. He blinks a position nearby, and he's invis for the next couple of seconds, so you might not catch him using these AOE abilities.

Yes. Most annoying thing: playing as blitzcrank and rocket grabbing one of teemo's mushrooms.

I'm pretty sure Teemo's mushrooms can no longer be pulled towards you, though they'll still block a grab during the brief period before they stealth.