Does egg make the cookies rise(puffy)?
The texture of cookies is a complex interaction of many factors, including the size of the cookie, the temperature of the oven, the amount of leavening, the way the fat is treated, and so on.

The flat, thin cookie with the cracks on top that you describe and desire is achieved by slightly under-baking the cookies, allowing them to rise, and then taking them out of the oven.  They then cool and deflate, causing the "cracks" and thinness.

The most likely culprit is that you are over-baking your cookies, and they are setting in the oven while fully domed up.  They would be fine cookies, but more of a crispy style.

A secondary, and related, contributing factor would be having the oven temperature too high.  If you don't already have one, an oven thermometer is an inexpensive and helpful investment to make sure you are baking at the desired temperature, as many ovens are off by a fair margin.

Note also that these are quite large cookies, baked with a "large" scoop, only six per tray.  That helps achieve the fallen state.  Don't make your cookies too small, if you are looking for this texture.

To answer your sub-questions:


Creaming cannot apply to a melted butter cookie, as it requires a solid but plastic fat to incorporate air into the cookie
The egg contributes structure (from protein), tenderness, and usually a fairly considerable proportion of the overall water in the cookie dough
In this recipe, the baking soda is primarily present to promote browning, as there is very little acid for it to react with other than from the molasses in the brown sugar (it will leaven a little).   Since these are "fallen" cookies, it is harder to get good browning and therefore flavor development.  The soda is to promote that.