This is very individual, I've seen people here say they have anything between extreme empathy to nearly none at all.
Personally, I feel sick when it comes to the suffering of animals, and I have had the same kind of reaction as OP describes.
When it comes to people though, my empathy is selective. I need to personally care for or identify with them in order to feel it. Otherwise, while I might understand why they are sad, happy or upset, it doesn't affect me.
lavellan, I'm glad more people than just me care about the humble worm. I once found a worm that had been run over by a car. The last half of its body was a mess and it couldn't move away from it. Knowing that worms can survive being cut in half, I cut off the destroyed part and carried the worm to a flower bed. I still think of it occasionally and hope it lived for however long worms live. If we had a garden I would have taken it home and placed it there, but we don't.