Let danger run on `pull_request_target` (#34370)

Summary:
This is a nit. Did some cleanups on our Danger setup.
The most substantial work is:
- Moved over danger to run on `pull_request_target`. This allow us to re-use the Github Action access token without having to provide one.
- Fixed an issue with the `Pick Request` label which was not applied correctly.
- Removed the danger_id as we were not passing it correctly.

## Changelog

[Internal] - Let danger run on `pull_request_target`

Pull Request resolved: https://github.com/facebook/react-native/pull/34370

Test Plan: Tested locally with `yarn danger pr https://github.com/facebook/react-native/pull/34197`

Reviewed By: cipolleschi

Differential Revision: D38533144

Pulled By: cortinico

fbshipit-source-id: 178ce411eb956870563c4d51719ed4dae1f1536d