Yes, she is 3. They are just very curious about everything. Our grandson is 3 and just loves to open everything. So our house is very child proofed and you really can't leave him along for very long as he will find something to do.
In terms of hitting people, she needs to be taught that is not ok. One way to do that is to get the book, "Hands are not for hitting". It is aimed at kids from 4 to 7 but should also work well at her age. It is meant to be read aloud and practiced or referred to. You can read about it here - https://www.amazon.com/Hands-Hitting-Ages-Best-Behavior/dp/1575420775
Oh, and other books that might be helpful are listed on this same link.
So a lot of what she is doing is what I would call kind of normal, but I would need more information. Even the spitting is kind of a new, fun, thing to do. You are going to find out that everytime you stop one thing - they will find another. Which is also what makes them so much fun to be around.
Does she go to any kind of a preschool where she might be picking up some of these things - like the spitting?