If you really are using craft felt it cannot be washed, you may want to use flannel instead. As for hanging it to the wall try using the Command Strips. It may take 4 or 5 to hold it but your wall won't get messed up. As for the hangers on the foam core you can fashion them from just about anything (example large metal paper clips).

design wall
 

My wall is foam board nailed to the entire wall. A cheap flannel backed table cloth sprayed with Removable tacky glue and put on the wall been up there two years no problems. If it gets too bad I can replace it easily. If it ever comes off the wall the 8 or 10 nail holes shouldn't be too hard to repair.

I used Command Strip hooks at 2' intervals high on the wall (I am 6' tall, so high is fine for me) and then used banker clips (those black ones with silver wire handles) on each hook. I can hang a small quilt top or the flannel and it holds plenty of weight, doesn't mar the fabric, portable and can be left up or put away as needed.

Make a pillow case & insert the foam board...attach loops or a rod pocket to the top & hang from a curtain rod.