Send Email through a widget form

There currently is no API to send email via a widget.  If you wish to do this you will need to write a custom widget extension or REST endpoint to call and then you can use the in-process API to do this.  I suggest reviewing the developer training in Documentation as well as the API docs.

Can we use Telligent SMTP server in our javascript code, instead of calling an external SMTP solution? If yes, which would be this SMTP address?

You don't provide an SMTP server to the API, its all encapsulated.   community.telligent.com/.../sendemail-in-process-api-service

You don't provide an SMTP server to the API, its all encapsulated.   community.telligent.com/.../sendemail-in-process-api-service