Hey Brian -
Yes, one can set parameter values the same way you can play with filter values on your embed links.
In the second scenario, we do prompt users to logon, and we do it on purpose. Just because they have the "Share" link doesn't tell Tableau Server that a person is acually allowed to see what is on the other side of that link.
The idea behind embedding is to give people an easy way to drop Tableau content into their branded portals or websites without exposing the Tableau portal itself - For example, you might want to embed a single dashboard on a SharePoint page, but not allow people to surf through the whole Tableau Server willy-nilly.
Can you give us a better idea of what your goals are? Based on the context of your message, I'm guessing you want to allow "your" Website to authenticate the user, and when a user hits a page on the Website afterwards, you want the Tableau viz to show automatically, assuming that if the Website OK'd the user to hit the page, then it's OK for Tableau to show it. If you would, give us your "perfect" scenario, ala:
1. User logs into my website
2. User navigates to to a page using a menu on my website called "dashboard"
3. User sees Tableau dashboard without having to login inside the page.