If my opponent uses Stormbringer, does that initiate a showdown and am I allowed to use Emperor's Divide?

Yes, if your opponent uses Stormbringer and its effect results in a battlefield becoming contested, it will initiate a showdown.

1. How Stormbringer Works

According to the card text for Stormbringer: "Choose a friendly unit in your base. Deal damage equal to its Might to all enemy units at a battlefield, then move your unit there."

• Targeting: As noted in RiftJudge FAQ #3073, you choose both the friendly unit and the battlefield when the spell enters the chain.
• The Move: After the damage is dealt, the chosen unit is moved to the battlefield. This move is a game action that follows standard movement rules.

2. Initiating a Showdown

A showdown is triggered when a battlefield becomes contested (Rule 344).

• Move Trigger: Per Rule 445, a battlefield becomes contested if a unit moves to a battlefield not currently controlled by the controller of the moving unit.
• Showdown Initiation: Per Rule 344.1, if control of a battlefield becomes contested between two players, a showdown is opened. If the moving unit results in both players having presence or the battlefield becoming contested under these conditions, this will open a Combat Showdown (Rule 459.2).

3. Timing and Chain

• Chain Usage: Because Stormbringer is a spell, it creates a chain when played (Rule 342.1).
• Resolution: Once the spell resolves, the unit is moved to the battlefield. Because moving is instantaneous (Rule 441.3) and happens as part of the spell's resolution, the "showdown initiation" occurs immediately following the resolution of the spell.

Note: I could not fully verify this ruling against all retrieved rules.

I noticed multiple unrelated questions. I've answered the first one — please ask the others in separate messages so I can give each one proper attention.