They don't have to be wireless, just depends on what the device supports.
Fire TV Stick is wireless and the regular Fire Stick has an ethernet port. My Blu-ray is both wireless and has ethernet. Most TVs are wireless.
I would say wireless is the way to go just because it's nice to have fewer wires hanging out the back. If performance is a concern, or if you have a lot of wireless devices already, you might want to find something that has an ethernet port. Just keep in mind you'll have to run it to your router or switch somehow.
My suggestion is to get the Fire TV Stick for $40. It will come already programmed with your Prime account so you can just plug it in and go. This will get you in the game the quickest. Then, when you redo your den you can move the Fire TV Stick to another TV in your house.