Hello fellas =)

Just wanted to ask this question:

Is it better to use sub domain of your website domain, or to use sub domain of a separate domain dedicated for emailing?

Even better:
If the website domain has bad reputation - and that we want to help restore the reputation - is it better to re-segment this damaged domain into subdomain, or should we simply go for a separate domain that isn't linked to a website, and create sub domains there?