Is Gmail totally transparent?

I am thinking of switching from Regular DreamHost Email to Gmail because sometimes DreamHost is down.

My concerns are:
will people (receving our emails) know that it is coming from or passing through Gmail?
Can our email address remain the same as with DreamHost?
Is there any kind of advertising added to the emails we send?

Absolutely not. Headers will always give away the true path of mail. Then again, Regular Dreamhost Email isn’t totally transparent, either, for the same reason.

It will be the same address. No advertising goes out. It’s a standard Gmail account, but they swap out for