Can Zed’s Shadow Clones conquer an open battlefield with zero might? Can it also hold a battlefield alone with zero might?

Yes to both — a unit's Might has nothing to do with Control, Conquering, or Holding.

A 0-Might Shadow Clone is still a unit (Rule 187.11), and control is established purely by having your units present at a battlefield at the end of a Showdown. If a clone is the only unit at an open battlefield when the Non-Combat Showdown closes, you establish Control and Conquer it for 1 point if you haven't already scored that battlefield this turn (CR 348.2.a, 471.1). Similarly, it can Hold: a clone sitting there alone during your Beginning Phase keeps you in Control and nets the Hold point (CR 190.4.a, 471.2).

The real vulnerability is combat, not Might: in a Combat Showdown the clone adds 0 damage to your pool and dies to any non-zero damage assigned to it (CR 467.2.c.2). And the normal once-per-battlefield-per-turn scoring limit still applies.