Your question has so many variables that could happen it would be next to impossible to cover all of them. The look back rule is not in effect until the batter/runner reaches 1st base. Runner at 3rd can stand off the bag until the batter/runner reaches 1st, but then must immediately choose to advance or return. If the pitcher makes a play on one of the runners, the look back rule is off.
As long as the runner on 3rd does not violate the look back rule, yes they can steal home.
I guess I always thought it was a dead ball once the batter was walked. I have never seen a steal from a runner other than the batter before (on a walk) and this had people in both stands thinking the runner on 3rd should have stayed put.
My answer went poof, so if this is a double post, forgive me.
Your local league may have a rule that you can't steal home. Other, than that, be ready to steal on a wild pitch, passed ball or if the catcher overthrows the pitcher.
You could have a delayed steal, when the catcher releases the ball, too.