yes, once an item is felted, it becomes a solid piece of fabric and can be cut. i used like an exacto knife kinda deal so i could TRY to make a circle for the wheel, try being the operative.
it did come out okay, though, and then i just did like a runner stitch i guess around the circle and square i had cut more for embellishment, but that would also prevent any potential fraying. the pattern from knit 1 actually has you bind off stitches and pick them up again to make the space for the screen, but did not have the space for the wheel. the pattern from KP for the felted cozy was just obnoxious and i hated it, but that's where i got the cutting idea from. i kinda did a combination of the two patterns.
as for the felting, this would be a perfect first felting project for you because of the size of the item. it's felted in the sink and with the KP sock yarn felted VERY quickly (and was SO fun!) there WAS a chained lanyard cord for it, but it went down the drain of the bathroom sink during the felting process and i haven't remade it yet.