Entomb then goes to the graveyard and is exiled, correct?
Negative.
You are correct that Ulamog will hit the graveyard before Entomb does. However, Ulamog's ability cannot go on the stack and resolve until Entomb has finished resolving, and as the last part of doing that, you put it into your graveyard. Therefore, Ulamog's trigger will shuffle it in along with the other contents of your graveyard, and (assuming nothing else happens in response to the Bog trigger,) nothing will get exiled.
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No, Entomb will be shuffled in with the rest of your graveyard. Ulamog's ability is a triggered ability that uses the stack, so it only goes on the stack after Entomb is done resolving and is in the graveyard. The graveyard is only shuffled when Ulamog's ability resolves, after players have had the occasion to respond to it.
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In response I cast Entomb, searching for Ulamog, the Infinite Gyre, my graveyard and Ulamog are shuffled into my library.
Entomb then goes to the graveyard and is exiled, correct?
Negative.
You are correct that Ulamog will hit the graveyard before Entomb does. However, Ulamog's ability cannot go on the stack and resolve until Entomb has finished resolving, and as the last part of doing that, you put it into your graveyard. Therefore, Ulamog's trigger will shuffle it in along with the other contents of your graveyard, and (assuming nothing else happens in response to the Bog trigger,) nothing will get exiled.
Currently Playing:
Legacy: Something U/W Controlish
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Hypercube! A New EDH Deck Every Week(ish)!
Makes sense when you two lay it out, but gets confusing.