The mail receiver is independent of Discourse and email sending, it just acts as a simple email server configured only to receive emails, using the Discourse API to push those emails in to Discourse.
The only thing I can think of in setting it up that might affect sending is setting reply by email address, though that should just set Reply-To on the outbound emails and not affect the sender address.
Just to confirm, it sounds like you have these (among the rest) in app.yml:
DISCOURSE_SMTP_USER_NAME: admin@tasat.org
DISCOURSE_NOTIFICATION_EMAIL: admin@tasat.org
And this in reply by email address:
replies+%{reply_key}@forum.tasat.org
Is that right?
If so, do email notifications that don’t accept replies still work? I believe the email verification is an example of that, so you could try creating an account and see if that email is sent successfully.
With that arrangement, what should happen for emails that can accept replies is that the sent email uses:
From: admin@tasat.org (also sender address)
Reply-To: replies+abc123@forum.tasat.org
Typically Reply-To is not treated as part of the sender information, it just provides a default To address to use when people reply, but perhaps Hostinger do treat it as such. You might be able to add a sending alias for replies@forum.tasat.org.