Can I send email as admin@mysite.com through this (actual site is hosted on Linode)? I've been using mandrillapp for this purpose, but it is going paid-only so I'm looking for alternatives.
However, if you need to send thousands of messages a day, you are probably sending automated mails, transactional or newsletters. Migadu is not intended for that, just as Gmail isn't.
You don't need Mailgun for that either, you can setup your own server to send mails on DO, or go for Amazon SES (Simple Email Service), cost is negligible.