Also in the US. Doctor visits for me come out of sick leave, a yearly-recycling collection of days that I'm allowed to take out for reasons of sickness. I have a separate set of days for vacation time.
As for funerals in particular, I haven't run in to this situation, so I don't know what I would be doing.
In Germany we have no sick days. If we are sick, we stay home. If we are sick for 3+ days we need a note from the doctor. But this does not affect our vacation time at all.
OK I'll provide a few tips as you're obviously doing it wrong ;)
The vacation days are what you should be using for stuff like doctor's visits and sickness. The sick days are to be kept forever...or an actual vacation. Normally on termination you get reimbursed for sick days. The vacation days are gone however.
Hmm... I suppose policies may vary, but in my experience exactly the opposite tends to be true. Sick days and personal days vanish at the end of the year while vacation days generally accumulate (sometimes with a cap, though). It's fairly common for people with separate pools to use "sick" days to just take a day off to recharge or run errands. (Although, technically, employers often have the right to demand a doctor's note, but seldom actually do so.)
Many companies have a policy for bereavement leave. The amount may depend on your relationship to the deceased. For example, it might be a week for a parent, spouse, or child; 3 days for a grandparent; etc.
As for funerals in particular, I haven't run in to this situation, so I don't know what I would be doing.