Can you react to triggers during the beginning phase? For example, can a hidden Teemo at a battlefield react to the death of a Sprite Token in order to not be discarded?

Yes, you can react to triggered abilities during the Beginning Phase, and you can absolutely use a hidden Teemo to save yourself from discarding.

According to Rule 383.3, when a condition for a triggered ability is met, it behaves like an activated ability and is placed on the chain. This means the Temporary keyword, which triggers at the start of your Beginning Phase (Rule 816.1.c), creates a valid window for reactions.

The Sequence

When a Sprite token (which has the Temporary keyword) is at a battlefield with a hidden unit like Teemo, the interaction follows this sequence:

1. Trigger Phase: At the start of your Beginning Phase, the Temporary ability of the Sprite token triggers.
2. Chain Starts: The "kill this" trigger goes on the chain.
3. Reaction Window: You now have an open window to play a [Reaction]. Because playing a hidden card is at reaction speed (Rule 811.6), you can flip your hidden Teemo face-up.
4. Resolve Reaction: Your Teemo is revealed and enters the board.
5. Resolve Trigger: With the chain now resolving (LIFO - Last In, First Out), the Sprite's "kill this" trigger resolves. The Sprite token is discarded, but because Teemo is now present at that battlefield, you maintain control of it.

By revealing Teemo in response to the trigger, you ensure you do not lose control of the battlefield, preventing the scenario where you would have been left with no units and thus potentially discarded.

This is based on multiple RiftJudge FAQs (e.g., FAQ #659, #5374) which confirm that Temporary is a triggered ability that uses the chain, allowing you to react by revealing hidden cards.