Custom API integration to Jira Cloud

Hello,

the customer has a self-made ticket system with an API.
The API accepts creation and updates of Tickets in a specific format.
The API can send creation and updates of Tickets to a configurable endpoint. The payload has a specific form.

What options do I have to integrate this with Jira Cloud?

I saw for outbound messages we could use Jira Automation and send RET requests.

Is oAuth supported for these REST Messages?
Are there other, maybe better, options than automation?

For inbound I saw that we can use Webhooks. But it seemed quite cumbersome to do all the mappings in the automation engine, as there seems to be no scripting option available.

Are there better options available here?
Or can the automation engine be expanded to support scripting or field-value-mappings?

Thank you for your help!