docker, ci: Make e2e tests less flaky

[ix5]

Checklist

• All new and existing tests are passing
• (If adding features:) I have added tests to cover my changes
• (If docs changes needed:) I have updated the documentation accordingly.
• I have added an entry to CHANGES.rst because this is a user-facing change or an important bugfix
• I have written proper commit message(s)
  

What changes does this Pull Request introduce?

• Check whether the comment really got created and otherwise fail faster in the integration test before PUT request
• Improve a few documentation comments

Why is this necessary?

End-to-end tests have been flaky when run in the dockerized environment. This seems to fix it locally for me, using the following steps:

make docker
docker compose up -d
make docker-js-integration