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

.github/workflows/danger_pr.yml

@@ -1,24 +1,28 @@
 name: Run Danger on PR
 
 on:
-  pull_request:
+  pull_request_target:
     types: [opened, edited, reopened, synchronize]
 
 permissions:
-  contents: read
+  actions: write
+  checks: write
+  contents: write
+  issues: write
+  pull-requests: write
+  statuses: write
 
 jobs:
   danger:
     runs-on: ubuntu-latest
     steps:
       - uses: actions/checkout@v3
-      - run: yarn install
+      - name: Run Yarn Install on Root
+        run: yarn install
         working-directory: .
-      - run: yarn install
+      - name: Run Yarn Install inside Bots
+        run: yarn install
         working-directory: bots
       - name: Danger
-        run: DANGER_GITHUB_API_TOKEN="$PUBLIC_PULLBOT_GITHUB_TOKEN_A""$PUBLIC_PULLBOT_GITHUB_TOKEN_B" yarn danger ci --use-github-checks --failOnErrors --id danger_pr
+        run: yarn danger ci --use-github-checks --failOnErrors
         working-directory: bots
-        env:
-          PUBLIC_PULLBOT_GITHUB_TOKEN_A: a6edf8e8d40ce4e8b11a
-          PUBLIC_PULLBOT_GITHUB_TOKEN_B: 150e1341f4dd9c944d2a