SurveyMonkey custom variables in embedded survey

I'm trying to use a SurveyMonkey embedded survey in our website. I know we can pass custom variables such as username in the url using a Web Link Collector. However I don't see any articles about passing custom variables using an embedded survey. Is it even possible?

Or maybe someone could guide me how to achieve my goal with SurveyMonkey? Here is the scenario I would like to achieve.

1. We would like to popup a survey when a user logs in.
2. We would like to get back which survey and when it was complete for each user from SurveyMonkey.
3. We would like to store the data in the database we get back and determine to popup the same survey when a user logs back in or not.
4. Repeat this every year.

Can SurveyMonkey do this?

I spoke to the SurveyMonkey help desk, and I was told that it is not possible to use custom variables in their embedded survey.