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If Im defending with Raider Visper, who is untapped, and my opponent plays EMP Wave, do I get the option of discarding 3 green pips to force them to discard 3 cards? If theyre matching, do I get to draw three? Also, EMP Wave would always make it the turn of the person who didnt play it, correct? Or if Im attacking and play EMP Wave do I get to swing with a character that I just tapped?

No, a Tap ability is something you use on your turn, not whenever the character becomes tapped. If you play EMP Wave on your turn, all characters will be tapped. Your turn continues, and you can play an upgrade, flip a character, etc. At the end of your turn, all characters will untap, and your opponents turn will begin.
Source: Wizards of the Coast (2019-05-02)    【Link to Ruling】
Tags: Tap