My DS went about a couple of years ago with a group.

Israel is more secular than you think. The Wailing Wall separates the women from the men. I think it is out of respect for the ultra orthodox sect that also lives in Jerusalem.

So you can take a scarf but I have not heard of any instance where you are required to wear it.

I hope you get to go on an archeological dig. That is one place I would love to visit. Entire neighborhoods completely buried by the sands of time.