Did the Tree of Knowledge have anything to do with sin?

Yes, eating of it was forbidden.

Was the first Sin that we ate of the tree or that we "hid" from God?

The eating occurred first.

Is there a lesson in shame? Makes me think of damnation.

Yes, the disobedience led to separation from God because the man and woman felt shame. They knew they disobeyed.

What causes shame?

A Bible Dictionary is a great resource to unpack the meaning in the ancient context. Here’s some of what Lexham has.

SHAME Feelings associated with (but not limited to) failure, public exposure, disgrace, embarrassment, social rejection, ridicule, and dishonor. How Was Shame Viewed in the Ancient World? Generally speaking, shame was a core value in the biblical world (Pilch, Introducing, 49). Greeks, Romans, and Judeans all considered shame to play a pivotal role in their cultures (Neyrey, “Shame of the Cross,” 115). This perspective can be difficult to understand in modern societies, where shame is largely regarded as a private problem. “Guilt” might better describe the corresponding modern value (Pilch, Introducing, 49).