Google Apps in DH terms is more about using existing Google Applications like Calendar, Mail, Documents, and Start Page in conjunction with your existing DH site that is hosted either on DH or on Google.
Applications like Calendar is meant to give you and your website maintainers (or whomever you want to give access to) a place to schedule events or to let your visitors know what you're up to.
For instance, on a music artist's website, the Calendar function could be used to schedule out music release dates and concert dates.
For a company's website that's dedicated to its employees, the Calendar can be used to display and organize company-specific and company-viewable-only events.
In relation to this, the Google Docs application can allow for easy filesharing between employees at the company's employee website.
The applications can be accessed through subdomains - calendar.website.com for example - or the URL and location can be changed through both DH's control panel and Google Apps control panel.
It's a CNAME redirect, so calendar.website.com == google's url for the calendar as Google is hosting the application, I believe, but I think it might be possible to cloak the google address with the calendar.website.com.
As for using Gmail:
Here's what my detailed email header looks like when I send out email using Gmail.
date Tue, Jul 8, 2008 at 7:41 PM
subject Test email
I'm sure if you use an email client like Thunderbird, other information is made available, but when using the web based email client, that's what the detailed header shows.
I'm sure I've missed some questions, but I hope I've answered a few and didn't confuse you any further.