Regarding Zed Legend, can I freely banish cards on my own to empower without needing another card with a banish effect, similar to how destructive can Recycle to gain +1 Might?

No — you can't banish "for free" to empower Zed. Banishing is a Limited Action: you may only banish cards when a game effect directs you to (Rule 427.4). There's no standalone "banish" cost or action like Vi's Recycle cost, and Master of Shadows itself doesn't have one — its activated ability is "Disempower me, [T]: Discard 1, then draw 1," and discarding is not banishing (Rule 427.2.b), so it won't trigger the empower.

To empower, you need an actual card effect that banishes a card you own — e.g., a Shadow Clone's attack trigger ("When I attack, you may banish a unit from your trash"), Death Mark's Flow, or other banish effects. Unlike Vi, whose ability lets you pay Recycle as a cost, there's no equivalent self-serve banish here.