How to populate the fields values from my database into a document template and send it to different email addresses?

Hello, @Max 

Welcome to the Docusign Community!

Using a Template (not via API, but Web) you can populates fields in an envelope reading a .CSV file. That’s possible through the Bulk Send solution.  Sending via Bulk Send reads a .CSV to the the recipients names, emails and any other field present in the Template.

Perhaps you can create a template then via API try to send envelopes using a Bulk Send operation.

In the Github there is something may shed a light for you:
https://github.com/lingkio/docusign-connect

Other examples using Bulk Send:
https://github.com/search?q=docusign+bulk&type=code

I hope that helps.

Alexandre