Your finger condition does indeed sound like eczema, but are you sure that when the "skin flakes and peels away and then it cracks and bleeds a bit," it isn't because you're helping it (by picking and scratching)? If you are, I advise you to stop, keep the skin moist (vaseline, etc.), and ask a skin doctor to diagnose you properly.
Eyelash loss can certainly be alopecial areata. It can also be caused by the habit of plucking the lashes. No one knows the cause of alopecia, but I can't think of any role for strep in either of your conditions. Strep of the skin causes infected follicles or impetigo.
Take care.
Dr. Rockoff