Cooling on the hot pan, versus removing immediately to wire racks, is a
matter of which cookie, and your own preferences and acquired
technique. Sorry, no one real answer here.

If tyou have a cooki recipe that rises, and you want the inside to stay
soft but the bottoms to brown and even get crispy, then leave for a
while on the hot pan. If you cook them to desired consistency, then
remove to the racks immediately.

As to storage, again that depends on the cookie. Some of my oatmeal
and peanut butter cookies will last in sealed containers for a couple
of weeks, just fine. Thinner, cruisper, more butter based cookies only
a few days before they are dried out.

and oh yes -- butter and lard are animal fat, marjarine and Crisc are
vegetable. fats. The first two generall melt soon in baking and make
cookies spread out and more crispy (but better tasting . The
marjarine or Crisco have more water content and aid in (some) cookie
recipes to rise larger and puffier. They look better, but to me are
not as tasty. Your choice.