Hamlet Song Summary Act 3

The transcript describes a scene where a character's boyfriend has gone mad, and she seeks comfort from her father. Meanwhile, Hamlet is indifferent, focused on Claudius's prayer, and talking to imaginary people. His mother thinks he's crazy, but after he reveals the truth, she begins her path to redemption. The transcript ends with a hope that Hamlet doesn't ruin everything.

What is Hamlet doing while Claudius is making his prayer?

What does the phrase 'on the road to redemption' imply about Hamlet's mother?