If I set the display name to something like "Bob Barker" email@example.com - would that be considered as spam?
Whereas "No-Reply" firstname.lastname@example.org would not be considered as spam?
Note: Where domain.com is my production domain name.
I would set it to the name of your company or service.
Depending on the law in your country, you may have a legal requirement to include additional information about your company on any email you send, such as your registered company number, company name, registered office address, etc. You should also provide a means by which people can opt-out of future emails.
I would say that you could make the email email@example.com and then just not accept emails at that address. Another option would be to give the email a Return-path of firstname.lastname@example.org. I make this suggestion as some email clients may be configured to not show the display name at all. I know that if I see a conflicting display name like that, I would think it to be spam.
You may want to remember that there are other reasons that your email by be considered spam. For instance, Hotmail does a reverse DNS lookup on email addresses. If your site is sharing an IP as a virtual host, you may have problems.