If Rayami, First of the Fallen dies does its ability exile itself?
Relevant card text:
If a nontoken creature would die, exile that card with a blood counter on it instead.
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The first paragraph creates a replacement effect. A replacement effect changes what happens as it happens, rather than reacting to it like a triggered ability does.
When Rayami, First of the Fallen would die (would be moved to the graveyard from the battlefield), it must necessarily be on the battlefield. As such, it is subject to its own ability.
Put differently, a game of Magic is a sequence of discrete states. This means that a creature that dies is on the battlefield one instant, and in the graveyard (or exile) the next. It's never in between. That means that Rayami must be on the battlefield when you decide to where to move it (because it couldn't possibly be in the graveyard or exile already).
Of course, if Rayami is your Commander, you don't have to exile it; you could choose to move it to the Command Zone instead. In that situation, you'd have two replacement effects vying to change Rayami's destination, so you get to apply the effects in the order of your choice (though the order doesn't matter here).
Yes, Rayami, First of the Fallen will be exiled if it dies on the battlefield.
The first set of rules text on Rayami is a Replacement Effect. As per Rule 614 'Such effects watch for a particular event that would happen and completely or partially replace that event with a different event. They act like “shields” around whatever they’re affecting.'
This means the Replacement effects alter any events before they happen, so in this case before Raymani dies it's death is replaced with exile.
If you're playing with Rayami as your commander rule 903.9 means you still have the opportunity to send Rayami to the command zone.
614.1 Some continuous effects are replacement effects. Like prevention effects (see rule 615), replacement effects apply continuously as events happen—they aren’t locked in ahead of time. Such effects watch for a particular event that would happen and completely or partially replace that event with a different event. They act like “shields” around whatever they’re affecting.
903.9 If a commander would be exiled from anywhere or put into its owner’s hand, graveyard, or library from anywhere, its owner may put it into the command zone instead. This replacement effect may apply more than once to the same event. This is an exception to rule 614.5.