According to the SE.

That fails to answer the question- the Schroedinger equation isn't lorentz invariant (its not even fully Galilean invariant), so it can't tell you much about spacetime.

You can't just replace Schroedinger with Dirac or Klein-Gordon without leading inevitably to a field theory, which opens up new cans of worms.

Most formulations of MWI only require a "for all practical purposes" splitting. Like thermodynamic irreversibility.

A mental picture of thermodynamic irreversibility as a directed tree is indeed an appealing one. It becomes less appealing once your tree does not have any well-defined vertices or edges due to the issues I have outlined.

According to the SE.

The SE is non-relativistic, so it has absolutely nothing to say about propagation in spacetime. It does not even describe emission or absorption, an essential part of decoherence. You have to go fully monty QFT to talk about signal propagation, but no one talks about MWI in the context of QFT, as far as I know.

Merg[e]able states are not split worlds.

In MWI [eigen]states correspond to worlds, so I don't know what it means. I also don't know what you mean by mergeable states.

Do different worlds share the same spacetime and for how long?

Presumably.

This implies gravitational interaction between non-interacting worlds, so do they interact or don't they?

See Penrose on MW.

Feel free to quote... Just not his quantum consciousness speculations.