Lady of Clockwork

What happens when I choose an attached Equipment with Orianna, Lady of Clockwork?

It detaches from the object it is attached to when it becomes a unit. Although its equip ability can be activated and it can be targeted by Weaponmaster triggers, it cannot attach to a unit; nothing will happen when those abilities resolve.

In this rules update we've clarified that units cannot be attached to other game objects. If a unit were to be attached to a game object, it would need to stop being a unit first. If a non-unit game object becomes a unit, it will detach from anything it is attached to as it becomes a unit. If an effect would attempt to attach a unit to another game object without making it no longer a unit first, nothing happens.

Relevant rules: 143.5., 435.1.h., 718.6.

What happens when I choose a gear with Deploy located at a battlefield with Orianna, Lady of Clockwork?

That gear will become a unit-gear when the triggered ability resolves; if their controller doesn't control that battlefield, they will apply contested status to that battlefield, and a combat or showdown will be staged there in the subsequent cleanup. When the chain clears, that showdown or combat will open.

Relevant rules: 190.3.a., 323.7, 323.8., 323.11., 323.12.

What happens when I choose another gear with Orianna, Lady of Clockwork and the previous unit-gear stops being a unit? Will its attached Equipment fall off? If it's a deploy gear in base, do I get to move it?

Its attached Equipment will remain attached, and if it is a Deploy gear in base, it will remain in base.

Although units cannot be attached to objects, other objects have no such compunctions. Any attached Equipment that don't become units will remain attached to the gear even after it stops being a unit. Their Might bonus won't apply, since the gear doesn't have a Might, but they will continue to apply their effect text to the gear they are attached to.

The majority of gear with Deploy reference "this battlefield" or "hold here" in such a way that gaining a benefit from base is not possible. That being said, if you want to move it back to a battlefield you can either turn it into a unit again with Orianna, Lady of Clockwork or you can use movement effects that specify moving gear, such as Command: Attack.

Note that there is a small gap between finalizing the Orianna, Lady of Clockwork trigger causing the previous unit-gear to stop being a unit, and the triggered ability resolving and causing the new gear to become a unit. During that gap, neither gear is a unit unless it was already one prior to ever being chosen by Orianna, Lady of Clockwork.

Relevant rules: 137.3.b.