That dog is a precious ball of fuzz. I was at first uncertain what I was looking at in the first picture, since nothing really stood out in it except the nose and the fuzz and my old brain didn't pick up on the fact that the nose was sideways. All of the shots posted are great, once I figured out what I was looking at in the first shot. Closeups like this almost don't need backgrounds.
I have muslin, Kate, and backdrops made from plastic. For their creases, folds, creases, and wrinkles, I bought and use a hand held steamer intended for steaming clothing that has worked well to remove these un-desirable effects from my backdrops. It only holds about 1/2 drinking glass of water, so refills are frequent, but it works well, and is portable so I can take it on out of studio shoots easily, but it does need Mains power to operate. I always use distilled water, purchased in gallon size, from my local grocery store when I steam my backdrops, to avoid getting mineral stains on the backdrops. The plastic backdrops do require extra care, but the heat of the steam will be successful, if you are careful to use only enough steam to get the wrinkles or fold lines out. Too much and you may damage the backdrop.
This is the one that I bought, but it isn't for use on your Mains power. I live in the USA, so it's fine for me. Find something similar that works with your mains power and get it.
I like the Kate backdrops the most, because I like the velvet like face surface of them for their reflection free properties, but they do wrinkle too.
Get yourself a hand held steamer, use only distilled water in it, and I'm quite certain that you will be happy with how well it works for you.
Charley