Same is true for polar fleece / blizzard fleece (I use that instead of felt lots of times).
Good point!! I do a lot with felt and lint can accumulate very quickly!!*
I once embroidered on felt to make cosmetic bags with zippers, after the first design was stitched out I too had to clean the bobbin area. For the second one I just slid a piece of tearaway underneath the hoop and nothing hung around, didn't have to clean it. It works, try it.
Yes, cleaning is a must. Mine was fuzzy when I changed the bobbin too.
As for stabilizer, I use the plastic like wss on top, nothing on back. Works out fine.
Did you know that when you embroider on felt you do not have to use any tearaway