Does Gutter Palace go on the chain? And if so, can you react to it to summon units during the beginning phase?

Yes, Gutter Palace's "At the start of your Beginning Phase" ability is placed on the chain, and you can react to it.

Does Gutter Palace go on the chain?

Yes. According to Rule 383.3, when a triggered ability's condition is met, it is placed on the chain. Because the ability is placed on the chain, it creates a Closed State, which allows players to play cards or activate abilities with the [Reaction] keyword in response.

Can you react to it to summon units during the beginning phase?

You may activate abilities in response to the Gutter Palace trigger only if they have the [Reaction] keyword or if you have another effect that grants them [Reaction].

Sequence of Events:

1. Trigger: At the start of your Beginning Phase, if the conditions are met, Gutter Palace’s ability triggers and is placed on the chain.
2. Closed State: The chain is now active, creating a Closed State.
3. Reaction: Players may play [Reaction] cards or activate [Reaction] abilities.
4. Resolution: The chain resolves in reverse order (LIFO - Last In, First Out). If your reaction changes the game state so that the conditions for Gutter Palace are no longer met, the ability will resolve but fail to trigger its effect.

Note: I could not fully verify this ruling against the retrieved rules, specifically regarding the conditions for the triggered ability or the specific activated ability of Gutter Palace.