It says right on the package ‘great for baking’ and goes on to talk about baking a loaf in parchment and ease of transfer to the freezer.
I use this stuff for cookies (like right this moment) and it fills the house with smoke. Why? Isn’t it supposed to be fine in the oven? It says it is!
How much excess paper is there around the cookie sheet? I find that if there’s a significant amount, it starts to smoke and burn. Always make sure the paper is just barely covering the cookie sheet.
It’s just inside the edges of the sheet, so no excess!
You’ll probably have better luck with this in Cafe Society, where the bakers and cooks hang out.
Moving IMHO --> Cafe Society.
Are you using a fresh sheet for each batch? I find that if I use the same sheet for more than one batch it starts to burn.
How are the cookies doing? Is it possible your oven is not calibrated right, and you are baking at a much higher temperature that the paper is rated for? I had this problem with an oven once, set it for 350’, but it was heating to 500’! And make sure your paper isn’t touching the sides of the oven.
It doesn’t have to, it can quit any time it wants!
Get off its back!
Doh, a day late and a dollar short on “to look cool” 
Parchment is usually good to 400 or 450 degrees F. If you’re going higher than that, it could be a problem. I’ve also had problems in the toaster oven, with paper that works fine in the regular oven. Of course, you could just have lousy quality parchment.
To answer the questions:
No, I just use the sheet once per batch - and it leaves a film on the cookie sheet too. Is that normal?
The temperature is 350, and the cookies turned out great. It is an old stove though, so I’m wondering if, even though the cookies turned out fine (and all things I’ve cooked in this oven over the last two years have turned out fine), maybe the oven is cycling up and down in temperature or something?
It shouldn’t leave any residue on the pan. Are you sure you aren’t using wax paper? It will burn and leave a residue. Otherwise it sounds like you got a bad batch of parchment, or a low quality brand.