The "floppy disc" could be mylar - a polyester. -> fleece jackets made of recycled polyester
The disk itself will most likely be mylar, as you say, but the ferric oxide coating on each side, plus the attached steel spindle, may well make it uneconomical to recycle.
The outer case could be PVC - which is again recycleable.
On a 5 1/4" floppy disk, yes. On a 3.5" disk, the case is a harder plastic, possibly PE, PP, PS, or something similar, which should be recyclable.
The metal clip and the spring that pushes it closed will be two different steels.
My guess is you'd be lucky to find anywhere that'll take the disks whole, but if you can be bothered to take the disks to pieces, you may be able to recycle some of the parts.