I think it is...
An oven (aka a cooker, a range; usually with stove on top).
I make a song of silence, so sit and listen well.
It hums (fan +/ heater of the oven), and you have to wait for alarm that tells you the food is ready.
I hold in beasts of many kinds far from where they dwell.
You can prepare nutritious meals in an oven using edible parts of any and all animals (e.g. a cow from your farm, or a kangaroo from your cousin's farm in Australia), some of which are wild and scary. (Pheasants! = Evil bird-creatures jumping at you out of nowhere.)
I've taken some peculiar things, like artifacts and faces.
It can be used to dry pottery, crafts, etc. 'Peculiar' because you would not eat your ceramic bowl made in the pottery kiln (oven), but you can eat your oven-cooked lasagna. You use ovens to bake cookies, cakes, pizza etc, where the dough serves as a round flat surface symbolising a 2D face of a person when common features such as sugar eyes, nose and mouth are added/ drawn (pareidolia). There's a face-shaped cookie cutter and gingerbread man Christmas tradition.
I have thick skin and spent some time in a kitchen, of all places.
Now, thick skin could be referring to the 'armour shield' build of the oven to protect the house from melting around it; or that lots of wild animals that can be cooked in the oven, have thick hide to protect them from the cold, dehydration, and attacks from predators. Pork rinds are a crispy oven-roasted (or fried) thick chunks of pigs' skin. Ovens are most probably located in a kitchen part of the house, and it's not surprising but 'faces' needed a rhyme.
I am a place for some to get their training for the Show.
"Great British Bake Off" and all the cooking shows on TV. Some of the shows are competitive, so if a person wanted to win by cooking the best meat pie or the most delicious triple chocolate brownies, they would have to practice cooking and baking those meals in the oven before the competition.
If you can tell me who I am, then feel free to let me know.
Yes, feel free to let me know if you are who I think you are ;)