Domain mapping affect SEO of the domain?

I understand how to make domain mapping function properly (at the bare basics), this question is more SEO related...

Let's say I create a site (after a sub-domain, multi-site is enabled) and my site address is 'example.realwebsite.com'. If I want to enable the domain mapping plugin and set 'awesomewebsite.com' to point to this site, am I getting the best SEO links possible without changing the wordpress settings page for this site?

To clarify:
If I create a blog post, page, etc., after the domain mapping plugin is enabled, is it still using 'example.realwebsite.com/page1' as the page address? If it is, then when that page gets called (or clicked from a search engine) does it just redirect to 'awesomewebsite.com/page1'?

Or, is it truly publishing every page and post on the web (after I hit publish) as 'awesomewebsite.com/page1'?

I've manually gone into individual sites' "Edit" site setting page in the past and changed the domain address from 'example.realwebsite.com' to 'awesomewebsite.com' and then set up domain mapping (guarantying that all of my links would include this domain for SEO reasons)...but do I need to do this?

I'm just trying to more clearly understand how this plugin manipulates the multisite system and how to use it in the best way possible.

Thanks for your answers in advance,
Chais