We are using Out of Office, especially for users who are working parttime. The integration within the Office products are great for displaying the availability of the user, e.g. when writing an email message. Therefor, the email notification sent by Outlook/Exchange is unnecessary and even considered internal spam.
Is it possible to turn off the emails or to prevent sending them?
Apparently, in Outlook 2016 at least, if you don't enter any text (or clear what's there already), the auto-replies won't be sent, as a pop-up message confirms but the other stuff you mention should still work. You'd need to check this actually works though and is appropriate.