Is it considered child abandenment when a mother leaves her 3 week old baby with a friend of the family for a week so she can hang out with her boy friend, who is not the father?

Is it abandonment? No. Might it be seen as poor parenting skills. Yes. And for that you could have the child taken away from you or lose custody to the other parent.

Was the child well taken care of, feed, properly watched ?

You may use for grounds to try and get custody, is that what you are wanting