Chilled puff pastry vs frozen

Yes, is possible, and not uncommon. You may wish to dock it to prevent excessive rising.

Some pastries are made with puff pastry that is already baked separately, such as the famous Napolean or Mille-Feuille . For example, in this recipe (which uses commercial puff pastry), the instructions are:

Preheat oven to 425°F/220°C. Line 2 baking sheets with parchment paper or plain brown paper; set aside.

Unfold puff pastry sheet and trim edges to a 9-inch square. Cut pastry into nine 3-inch squares. Transfer pastry squares to the prepared baking sheets; prick pastry. Bake in a preheated oven for 18-23 minutes, or until golden. (Or bake according to package directions.) Carefully remove pastries from baking sheet. Cool on a rack.

I cannot endorse the plain brown paper idea, but the method is fairly standard.

Here is another example, from King Arthur flour, of croissants du patisserie, which is essentially puff pastry baked as a croissant (as opposed to croissants du boulanger, the baker's croissants, which are yeast raised as well as a laminated pastry).

No, you don't have to freeze it at all.

The recipe you linked is indeed for puff pastry. It is rather tedious to create it, so it is available in supermarkets as a pre-fabricated food, just like pizza dough and other doughs. But it doesn't have a long shelf life in the fridge, so it is sold frozen. The recipe assumes that you will buy it frozen, and warns you to let it come to room temperature before you start baking.

If you are making your own, the end product can be used in cookies immediately. Just pay attention to follow proper technique and work with a very cold dough and butter while making the puff pastry. The link you posted probably explains it - if not, search for better instructions.