Resolve deprecations in `API` tests

[boraberke]

This PR resolves internal warnings under tests/api_connexion.

Test Class	Issue	Resolution
TestGetConnection	Encountered non-JSON in extra field	Reformatted them as JSON
TestPatchConnection	Encountered non-JSON in extra field	Reformatted them as JSON
TestSetDagRunNote	The experimental REST API is deprecated	Removed this test. This test is already covered via another test test_with_auth_role_public_set
TestGetExtraLinks	BigQueryExecuteQueryOperator has been deprecated	Used CustomOperator instead
TestGetTaskInstancesBatch	get_accessible_dag_ids has been deprecated	Changed to get_auth_manager().get_permitted_dag_ids
TestConnectionSchema	Encountered non-JSON in extra field	Reformatted them as JSON

P.S. I was not sure if I needed to open a separate PR for the fix of TestGetTaskInstancesBatch, as it involves a change in the codebase. I included it here to address all API-related test deprecations at once. Let me know if I should separate it, and I will remove it from this PR and create a new one.

related: https://github.com/apache/airflow/issues/38642

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