Honestly, you really *should* see the vet on this one.
Reason why: the quick of the nail is *very* vascular - so not only can it bleed profusely but if there are actual splits, it means it's extremely vulnerable to infection. Also, the quicks are very innervated - which means PAIN.
Also also
, the nails should not be splitting as you're describing - so if it were me, I'd want thyroid checked and some perhaps some basic chem panels done, just to make sure all is okay.
The vet may want to wrap the paws and/or do boots for a while, btw.
Keep us posted...I'm curious to find out what's wrong here...