APGAR scoring is nothing more than a specific way to assess how a baby is doing at 1 minute of age and 5 minutes of age. 5 catagories of things (color, respirations, muscle tone, reflex irritability, and heart rate) are quickly assessed and assigned a score of 0, 1, or 2. Thus a baby born dead would have an APGAR of 0, while a baby who scores best on everything would be a 10. There is no "law" that says an APGAR score is done, but it is routine for all doctors and midwives to do this. Because all practitioners do this the same way, if you told another health care provider that this baby's APGAR scores were 3 and 8, that HCP would have a very good idea of exactly how the baby was doing at age 1 minute and 5 minutes.
Vitamin K is NOT put into babies eyes. Antibiotic ointment or drops (usually erythromycin) is usually put into the baby's eyes to prevent infection and blindness from undiagnosed gonorrhea or chlamydia. Most states have a law that requires this to be done by the time the baby is 2 hours old, although you have the right to refuse it.
Vitamin K is routinely given to newborns to help their blood to clot better, preventing hemorrhagic disease of the newborn. It is very rare. There are no laws mandating this, it is just routine. You have the right to refuse.
Read the fine print on EVERYTHING before you sign it! There are some things, like eye ointment, that have become 'standard of care', and it is assumed you have given permission unless you clearly state in writing that you refuse. Big things, like circumcision, are not likely to occur unless you have signed consent and had risks [somewhat] explained. However, I have known many moms who have refused things like Hepatitis B immunizations and vitamin K, or giving their baby formula from a bottle, and had these things given to their babies while they were in the newborn nursery anyway, against the parents wishes.
Ask lots of questions from your doc or midwife, your pediatrician, family doc, etc. Get answers you understand. It's your right.