Biz is equipped with Helmet. If suddenly BW is gone, does Helmet stay or go?

Stay, cause I don't think the "only equip to ..." is a continuous check.

This may be the wrong rule to quote but I think the answer is you keep the helmet.

"704.4 The uniqueness rule is only checked while resolving a character recruit or substitute effect, flipping a location, or attaching an equipment. It does not apply to any other modifiers that put an object into play or change of control of an object."

While we're at it... does Mutant Massacre stop Bizarro with Bizarro World?

Mutant Massacre:

To play, exhaust a Mutant you control.
Remove target stunned character from the game.

Ongoing: Your opponents can't recruit non-Army character cards with the same printed name and version as the character you removed.

So if Bizarro, Dark Mirror was removed by Mutant Massacre then whoever it was played against couldnt recruit another Bizarro, Dark Mirror until Mutant Massacres ongoing effect was removed some how.

Equip restrictions are checked as part of equipping/attaching. You can't stick Juggernaut's Helmet on anyone who isn't named Juggernaut. But as Em7add11 supposed, they are only checked when attaching. So losing BizWorld doesn't get rid of the Helmet, just like nuking a team-up can't get rid of any Utility Belts already in play.