Is it possible to integrate Mandrill with CiviCRM via their SMTP, or do I need to set up API via extension?
you can send via SMTP, so without any extension.
However, you will not handle events about bounces, so civicrm will not be able to update the contact to flag that the email is invalid.
So you will keep sending emails to the same invalid email, how mandrill will handle that (eg blacklist these emails, invoicing them as sent... up to ban you) is based on how many emails you have in that case and other factors.
Bottom line, if you don't want to risk them closing your account, you should install the extension and configure it properly, it's not a risk worthwhile taking