Hi kévin,
To get the DataMiner ID of the DMA your script is running on, you can use the Engine object in your API script, see following question: Automation – How to get the ID from the Agent running the automation script?

To answer your second question, the User-defined APIs logic will log failing API requests in one of the logfiles mentioned in here: Troubleshooting User-Defined APIs | DataMiner Docs depending on where it fails. Note that these logfiles aren't permanent and can be overwritten due to restarts or when a certain filesize is reached.
You could also log failures in your script yourself using the LogHelper | DataMiner Docs. If you use this approach, note that if you e.g. try to trigger an API using a wrong token, you won't be able to log this, as in this case your API script will never be started. For these kind of fails, you will always have to look at the logfiles mentioned earlier.