Hey there, yes you can just put the cage on the carpet, you just need to make sure you cover the carpet with something at first to prevent toilet accidents while you are litter training, and secondly to protect the carpet from bunny chewing. One thing to keep in mind with having a cage in your bedroom, is that rabbits are often very active in the early night and very early morning, so if you are a light sleeper, you may have trouble with this.
Dog x-pens/play pens are often recommended as cages because they are big and versatile. If you have a rabbit that isn’t inclined to jump or climb out, they work really well and offer a great amount of space. NIC or C&C I also recommend, I have one myself for my pair and it works great, offers plenty of room and is good for rabbits that might otherwise escape a play pen. I generally don’t recommend store bought cages unless you have a situation like Sarahthegemini where her buns are constantly free range and are never shut in their cages. Although some people do use the XL sizes of dog crates as cages for their buns with success.